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International Journal of Algebra and Computation
Vol. 20, No. 3 (2010) 357–380
c World Scientific Publishing Company
DOI: 10.1142/S0218196710005546
DADE’S INVARIANT CONJECTURE
FOR THE GENERAL UNITARY GROUP GU4 (q 2 )
IN DEFINING CHARACTERISTIC
JIANBEI AN
Department of Mathematics, University of Auckland
Private Bag 92019, Auckland, New Zealand
[email protected]
SHIH-CHANG HUANG
Department of Mathematics
National Cheng Kung University, Tainan, Taiwan
[email protected]
Received 13 March 2009
Revised 21 January 2010
Communicated by S. Margolis
In this paper, we verify Dade’s invariant conjecture for the general unitary group
GU4 (q 2 ), q a power of an odd prime p, in the defining characteristic p. Together with
the results in [3], this completes the proof of Dade’s invariant conjecture for the group
GU4 (q 2 ) in the defining characteristic.
Keywords: Modular representation of finite groups; Dade’s conjecture; general unitary
groups.
Mathematics Subject Classification: 20C20, 20C40
1. Introduction
Let G be a finite group and p a prime dividing the order of G. There are several
conjectures connecting the representation theory of G with the representation theory of certain p-local subgroups (i.e. the p-subgroups and their normalizers) of G.
For example, it seems to be true, that if P is a Sylow p-subgroup of G, then the
number of complex irreducible characters of G of degree coprime with p equals the
same number for the normalizer NG (P ).
This conjecture, called McKay conjecture [18], and its block-theoretic version due to Alperin [1] were generalized by various authors. In [17], Isaacs and
Navarro proposed a refinement of the McKay conjecture that deals with congruences of character degrees mod p. In a series of papers [9–11], Dade developed several conjectures expressing the number of complex irreducible characters
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with a fixed defect in a given p-block of G in terms of an alternating sum of
related values for p-blocks of certain p-local subgroups of G. In [10], Dade proved
that his (projective) conjecture implies the McKay conjecture. Motivated by the
Isaacs–Navarro conjecture [17], Uno [21] suggested a further refinement of Dade’s
conjecture.
In this paper, we show that Dade’s invariant conjecture holds for the general
unitary group GU4 (q 2 ) with q a power of an odd prime p, in the defining characteristic p. Note that in this case Uno’s invariant conjecture is equivalent to Dade’s
invariant conjecture. Together with the results in [3] this completes the proof of
Dade’s invariant conjecture for GU4 (q 2 ) in the defining characteristic.
The methods are similar to those in [2]. By a corollary of the Borel and Tits
theorem [6], the normalizers of radical p-chains of GU4 (q 2 ) are exactly the parabolic
subgroups. So we count characters of these chain normalizers which are fixed by
certain outer automorphisms. Our calculations are based on the character tables of
GU4 (q 2 ) and their parabolic subgroups which have been computed in [19].
This paper is organized as follows: in sec. 2, we fix notation and state the
Dade and Uno invariant conjectures in detail. In Sec. 3, we state and prove some
lemmas from elementary number theory which we use to count fixed points of
certain automorphisms of GU4 (q 2 ). In Sec. 4, we compute the fixed points of the
outer automorphisms of GU4 (q 2 ) on the irreducible characters of the parabolic
subgroups. In Sec. 5, we verify Dade’s invariant conjecture for GU4 (q 2 ), q = pn
odd, in the defining characteristic p. Details on irreducible characters and conjugacy
classes of GU4 (q 2 ) are summarized in tabular form in an appendix.
2. Dade’s Invariant Conjecture
Let R be a p-subgroup of a finite group G. Then R is radical if Op (N (R)) = R, where
Op (N (R)) is the largest normal p-subgroup of the normalizer N (R) := NG (R).
Denote by Irr(G) the set of all irreducible ordinary characters of G, and by Blk(G)
∈ Blk(G), and d is an integer, we denote by
the set of p-blocks. If H ≤ G, B
Irr(H, B, d) the set of characters χ ∈ Irr(H) satisfying d(χ) = d and b(χ)G = B
(in the sense of Brauer), where d(χ) = logp (|H|p ) − logp (χ(1)p ) is the p-defect of χ
and b(χ) is the block of H containing χ.
Given a p-subgroup chain C : P0 < P1 < · · · < Pn of G, define the length
|C| := n, Ck : P0 < P1 < · · · < Pk and
N (C) = NG (C) := NG (P0 ) ∩ NG (P1 ) ∩ · · · ∩ NG (Pn ).
The chain C is said to be radical if it satisfies the following two conditions:
(a) P0 = Op (G) and
(b) Pk = Op (N (Ck )) for 1 ≤ k ≤ n.
Denote by R = R(G) the set of all radical p-chains of G.
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Suppose 1 → G → E → E → 1 is an exact sequence, so that E is an extension
of G by E. Then E acts on R by conjugation. Given C ∈ R and ψ ∈ Irr(NG (C)),
let NE (C, ψ) be the stabilizer of (C, ψ) in E, and
NE (C, ψ) := NE (C, ψ)/NG (C).
∈ Blk(G), an integer d ≥ 0 and U ≤ E, let k(NG (C), B,
d, U ) be the number
For B
of characters in the set
d, U ) := {ψ ∈ Irr(NG (C), B,
d) | N (C, ψ) = U }.
Irr(NG (C), B,
E
Dade’s invariant conjecture can be stated as follows:
∈ Blk(G) with defect
Dade’s Invariant Conjecture ( [11]). If Op (G) = 1 and B
= 1, then
group D(B)
d, U ) = 0,
(−1)|C| k(NG (C), B,
C∈R/G
where R/G is a set of representatives for the G-orbits of R.
Let Aut(G) and Out(G) be the automorphism and outer automorphism groups
of G, respectively. We may suppose E = Out(G). If moreover, Out(G) is cyclic,
then we write
d, |U |) := k(NG (C), B,
d, U ).
k(NG (C), B,
Let H be a subgroup of G, ϕ ∈ Irr(H), and let r(ϕ) = rp (ϕ) be the integer
0 < r(ϕ) ≤ p − 1 such that the p -part (|H|/ϕ(1))p of |H|/ϕ(1) satisfies
|H|
≡ r(ϕ) mod p.
ϕ(1) p
Given 1 ≤ r < (p + 1)/2, let Irr(H, [r]) be the subset of Irr(H) consisting of those
∈ Blk(G), C ∈ R, an integer d ≥ 0 and
characters ϕ with r(ϕ) ≡ ±r mod p. For B
U ≤ E, we define
d, U, [r]) := Irr(NG (C), B,
d, U ) ∩ Irr(NG (C), [r])
Irr(NG (C), B,
d, U, [r]) := |Irr(NG (C), B,
d, U, [r])|. The following refinement of
and k(NG (C), B,
Dade’s conjecture is due to Uno.
∈
Uno’s Invariant Conjecture ([21, Conjecture 3.2]). If Op (G) = 1 and B
Blk(G) with defect group D(B) = 1, then for all integers d ≥ 0 and 1 ≤ r <
(p + 1)/2,
d, U, [r]) = 0.
(−1)|C| k(NG (C), B,
C∈R/G
Let q = pn be odd, G = GU4 (q 2 ) and K a parabolic subgroup of G. Then
r(ϕ) ≡ ±1 mod p for any ϕ ∈ Irr(K) (see the Tables A.2–A.5 in Appendix). It
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follows that
d, U, [1]) = Irr(NG (C), B,
d, U )
Irr(NG (C), B,
and radical chain C (see the proof in Sec. 5). Thus Uno’s invariant
for any block B
conjecture for G is equivalent to that of Dade. So we only verify Dade’s invariant
conjecture for G. Also notice that the Out(G) is cyclic and the Schur multiplier
of G is trivial. So the invariant conjecture for G is equivalent to the inductive
conjecture.
3. Notation and Lemmas from Elementary Number Theory
From now on, we always assume that p is an odd prime, n is a positive integer
and q = pn . We denote by N = {0, 1, 2, . . .} the set of natural numbers including
zero. In the next section, we will use the following lemmas, the first one is [2,
Lemma 3.1].
Lemma 3.1. Suppose m, n, a ∈ Z with m, n > 0. Then gcd(am −1, an −1) = |ad −1|
where d := gcd(m, n).
Lemma 3.2. Let t be a positive integer with t | 2n. Then the following hold.
2
if t | n,
(i) gcd(pt − 1, q + 1) = t/2
+1
p
(ii) gcd(pt − 1, q 2 − 1) =pt − 1.
(iii) gcd(pt + 1, q + 1) =
(iv) gcd(pt + 1, q 2 − 1) =
t
2
(v) gcd(p + 1, q + 1) =
2
if t n.
if 2t | n or t n,
t
p pt++1 1 ifift t| |nn,and 2t n.
22
pt + 1
if t n.
if t | n,
if t n.
Proof. (i) Suppose t | n. If d | pt − 1, q + 1, then d | q − 1 by Lemma 3.1, it follows
that d | gcd(q − 1, q + 1) = 2.
Suppose t n. There are k, tu , nu ∈ N with 2 | tu and 2 nu such that t = 2k · tu ,
k
k
n = 2k ·nu . Hence pt − 1 = (−p2 )tu − 1 and q + 1 = −((−p2 )nu − 1). So Lemma 3.1
k
k
k
implies gcd(pt − 1, q + 1) = gcd((−p2 )tu − 1, (−p2 )nu − 1) = |(−p2 )tu /2 − 1| =
pt/2 + 1.
(ii) is clear by Lemma 3.1.
(v) Suppose t | n. If d | pt + 1, q 2 + 1, then d | p2t − 1 and p2t − 1 | p2n − 1 = q 2 − 1,
so that d | gcd(q 2 − 1, q 2 + 1) = 2.
Suppose t n. There are k, tu , nu ∈ N with odd tu nu such that t = 2k+1 · tu ,
k+1
k+1
n = 2k · nu . By Lemma 3.1, pt + 1 = −((−p2 )tu − 1) | (−p2 )nu − 1. So
pt + 1 | q 2 + 1.
(iii) Suppose 2t | n. If d | pt + 1, q + 1, then d | p2t − 1 and so d | p3n − 1 as 2t | 3n.
Thus d | q 3 − 1, q 3 + 1 and d | gcd(q 3 + 1, q 3 − 1) = 2.
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Suppose t | n and 2t n. Then t, n have “the same 2-part”, i.e. there are
k, tu , nu ∈ N with odd tu nu such that t = 2k · tu , n = 2k · nu . Hence pt + 1 =
k
k
−((−p2 )tu − 1) and q + 1 = −((−p2 )nu − 1). So Lemma 3.1 implies gcd(pt + 1, q +
k
k
k
1) = gcd((−p2 )tu − 1, (−p2 )nu − 1) = |(−p2 )tu − 1| = pt + 1.
Suppose t n. If d | pt + 1, q + 1, then by (v), d | q 2 + 1, q + 1 and so d |
gcd(q 2 + 1, q 2 − 1) = 2.
(iv) Suppose t | n. Then 2t | 2n and pt + 1 | p2t − 1. Hence pt + 1 | p2n − 1 = q 2 − 1.
Suppose t n. If d | pt + 1, q 2 − 1, then by (v), d | q 2 + 1 and hence d |
gcd(q 2 + 1, q 2 − 1) = 2.
Lemma 3.3. Let t ∈ N\{0} with t | 2n. Define δ := 1 if t | n and δ :=
Then
pδt + 1 if 2t | n or t n,
gcd(pn−t + 1, q 2 − 1) =
2
if t | n and 2t n.
1
2
if t n.
Proof. Suppose 2t | n. There are k, tu , nu ∈ N with 2 tu and 2 | nu such that t =
k
k
2k ·tu , n = 2k ·nu . Hence pn−t + 1 = −((−p2 )nu −tu − 1) and q 2 − 1 = (−p2 )2nu − 1.
k
k
So Lemma 3.1 implies gcd(pn−t + 1, q 2 − 1) = gcd((−p2 )nu −tu − 1, (−p2 )2nu − 1) =
k
|(−p2 )tu − 1| = pt + 1.
Suppose t | n and 2t n. Let d = gcd(pn−t + 1, q 2 − 1), so that d | p2(n−t) − 1.
There is an odd nu ∈ N such that n = t · nu , n − t = t · (nu − 1) = 2t · nu2−1 . By
Lemma 3.1, it follows that gcd(p2(n−t) − 1, q 2 − 1) = p2t − 1, so d | p2t − 1 and
p2t ≡ 1 (mod d). Thus
0 ≡ pn−t + 1 = p2t·
nu −1
2
+ 1 ≡ 2 (mod d)
and d = 2 as d is even.
Suppose t n. There are k, tu , nu ∈ N with 2 | tu and 2 nu such that t = 2k · tu ,
k
k
n = 2k · nu . Hence pn−t + 1 = −((−p2 )nu −tu − 1) and q 2 − 1 = (−p2 )2nu − 1. So
k
k
Lemma 3.1 implies gcd(pn−t + 1, q 2 − 1) = gcd((−p2 )nu −tu − 1, (−p2 )2nu − 1) =
k
|(−p2 )tu /2 − 1| = pt/2 + 1.
Lemma 3.4. Let t, m be positive integers. Suppose t | n. If 2m | q + 1, then
2m | pt + 1.
Proof. Suppose 2t | n. There is nu ∈ N such that n = 2t·nu . Then q +1 = pn +1 =
p2tnu + 1 ≡ (±1)2tnu + 1 = 2 (mod 4). In particular, 4 q + 1 and 2 | q + 1. So
2 | pt + 1.
Suppose 2t n. There is an odd nu ∈ N such that n = t · nu . Then
q + 1 = ptnu + 1 = (pt + 1) · r,
(3.1)
where r = (pt(nu −1) − pt(nu −2) + · · · + 1). If 2m | q + 1, then by (3.1) and r is odd,
we get 2m | pt + 1.
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The following two lemmas follow from [2, Lemma 3.3] by replacing δ by 1 and
q − 1 by q + 1 (respectively q 2 − 1).
Lemma 3.5. Let t, m be positive integers. Suppose t | 2n and t n. If 2m | q + 1,
then 2m | pt − 1.
Lemma 3.6. Let t, m be positive integers. Suppose t | 2n and t n. If 2m | q 2 − 1,
then 2m | pt − 1.
4. Action of Automorphisms on Irreducible Characters
Let G = GU4 (q 2 ) be the general unitary group defined over a finite field with
q = pn elements (always assuming that p is odd). Let O = Out(G) and A = G O.
Then O = α and A = G α, where α is a field automorphism of order 2n. We
fix a Borel subgroup B and distinct maximal parabolic subgroups P and Q of G
containing B as in [19]. In particular, α stabilizes B, P and Q.
In this section, we determine the action of O on the irreducible characters of B,
P , Q and G. Our notation for the parameter sets of these groups is similar to that
of CHEVIE and is given in Table A.1 in the Appendix. The correspondence between
the CHEVIE notation and that of Nozawa is given in Tables A.2–A.5.
The first column of Table A.1 defines a name for the parameter set which
parameterizes those characters which are listed in the second column of the table.
The list of parameters in the third column of Table A.1 in the Appendix is of
the form
k = 0, . . . , n1 − 1
k = 0, . . . , n1 − 1 or
l = 0, . . . , n2 − 1
where the nj ’s are polynomials in q with integer coefficients. In the first case, the
parameter k can be substituted by an element of Z, but two parameters which differ
by an element of n1 Z yield the same character. In the second case, the parameter
vector (k, l) can be substituted by an element of Z × Z, but two parameter vectors
which differ by an element of n1 Z × n2 Z yield the same character. In other words,
k can be taken to be an element of Zn1 and (k, l) can be taken to be an element
of Zn1 × Zn2 . The groups Zn1 and Zn1 × Zn2 are also called character parameter
groups (see Subsec. 3.7 of the CHEVIE [14] manual). The next lines of Table A.1
list elements which have to be excluded from the character parameter group. The
remaining parameters are called admissible in the following. Different values of
admissible parameters may give the same character. The fourth column of Table A.1
defines an equivalence relation on the set of admissible parameters. If no equivalence
relation is listed we mean the identity relation. The parameter set is defined to be
the set of these equivalence classes. Finally, the last column of Table A.1 gives the
cardinality of the parameter set.
We consider the example P I3 in Table A.1. The character parameter group is
Zq2 −1 × Zq2 −1 . The parameter vectors (k, l) and (l, k) yield the same character and
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the equivalence class of (k, l) is {(k, l), (l, k)}. Hence, the characters
parameterized by the set
P I3
P β3 (k, l)
363
are
= {{(k, l), (l, k)} | (k, l) ∈ Zq2 −1 × Zq2 −1 , q 2 − 1 k − l}.
If we want to emphasize the dependence of a parameter set, say P I3 , from q we write
P I3 (q). Table A.1 does not give any detailed information about the parameter sets
G I2 , G I5 , G I6 , G I9 and G I10 , since we will not need an explicit knowledge of these
sets (note that these parameter sets parameterize the regular semisimple irreducible
characters of G). The data in Table A.1 is taken from [19].
The action of O = Out(G) on the conjugacy classes of elements of G, B, P and
Q induces an action of O on the sets Irr(G), Irr(B), Irr(P ) and Irr(Q) and then
an action on the parameter sets. Using the values of the irreducible characters of
G, B, P and Q on the classes listed in the last column of Tables A.2–A.5. we can
describe the action of O on the parameter sets.
For an O-set I and each subgroup H ≤ O let CI (H) denote the set of fixed points
of I under the action of H. In the following proposition we determine |CI (H)|
where I runs through all (disjoint) unions of parameter sets which are listed in
Table A.6 except for G I2 ∪ G I5 ∪ G I6 ∪ G I9 ∪ G I10 . This last union of parameter sets
will be treated separately since it requires different methods.
Proposition 4.1. Let G = GU4 (p2n ), t | 2n and I = G I2 ∪ G I5 ∪ G I6 ∪ G I9 ∪ G I10
be one of the (disjoint ) unions of parameter sets listed in Table A.6. If H = αt is
a subgroup of O, then the second and third columns of Table A.6 show the number
of fixed points |CI (H)| of I under the action of H.
Proof. We have to consider the parameter sets in Table A.6.
In each of the following cases, we have that the action of α on I is given by xα =
px for all x ∈ I using the character values on the classes listed in the last column
of Tables A.2–A.5. We demonstrate this for the parameter set I = P I3 ∪ P I8 . The
degrees in Table A.5 show that the P β3 (k, l)’s are the only irreducible characters of
P of degree q 2 + 1, so P β3 (k, l)α = P β3 (k , l ) for some {(k , l ), . . .} ∈ P I3 . We see
from the class representatives in Table VI-1 in [19] that α acts on the semisimple
conjugacy classes of P like the pth power map which implies that the values of
P β3 (k , l ) and P β3 (pk, pl) on the semisimple classes coincide. Then, the character
values of P β3 (k, l) (see the character Table VI-3 in [19]) imply that the values of
P β3 (k , l ) and P β3 (pk, pl) coincide on all classes, hence P β3 (k , l ) = P β3 (pk, pl) and
α
α
therefore P β3 (k, l) = P β3 (pk, pl). Similarly, P β8 (k) = P β8 (pk). Hence, xα = px
for all x ∈ I.
Let I ∈ {G I1 , G I12 , G I13 , G I14 , B I4 , B I8 , P I5 , P I9 , P I10 , Q I5 }. If k ∈ I, then k ∈
CI (H) if and only if (pt − 1)k ≡ 0 mod q + 1.
Suppose t | n. Then by Lemma 3.2(i), this is equivalent with 2k ≡ 0 mod q + 1.
So we get |CI (H)| = 2.
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Suppose t n. Then by Lemma 3.2(i), this is equivalent with (pt/2 + 1)k ≡ 0
mod q + 1. So we get CI (H) = {k ∈ I | k is a multiple of (q + 1)/(pt/2 + 1)} and
|CI (H)| = pt/2 + 1.
Let I ∈ {G I3 ∪ G I22 , G I4 ∪ G I20 , P I6 ∪ P I7 }. Then these unions of parameter sets
are isomorphic H-sets, so that we can assume I = G I3 ∪ G I22 . First, we compute
|C G I3 (H)|. Let
if i = 1,
{{k, −qk} ∈ C G I3 (H) | pt k ≡ k}
Ui :=
t
{{k, −qk} ∈ C G I3 (H) | p k ≡ −qk} if i = 2.
If x = {k, −qk} ∈ G I3 , then x ∈ U1 if and only if (pt − 1)k ≡ 0 mod q 2 − 1. By
2
−1
Lemma 3.2(ii), this is equivalent with k is a multiple of qpt −1
. So
if t | n,
(pt − 3)/2
|U1 | =
t
t/2
(p − p − 2)/2 if t n.
Suppose t | n. If x = {k, −qk} ∈ G I3 , then x ∈ U2 if and only if (pt + q)k ≡ 0
mod q 2 − 1. Then (pn−t + 1)k ≡ 0 mod q 2 − 1. By Lemma 3.3, this equivalent with
(pt + 1)k ≡ 0 mod q 2 − 1 if 2t | n, and 2k ≡ 0 mod q 2 − 1 if 2t n. So by the
definition of G I3 , we get
(pt − 1)/2 if 2t | n,
|U2 | =
0
if 2t n.
Suppose t n. If x = {k, −qk} ∈ G I3 , then x ∈ U2 if and only if (pt + q)k ≡ 0
mod q 2 − 1. Then (pn−t + 1)k ≡ 0 mod q 2 − 1. By Lemma 3.3, this is equivalent
with (pt/2 + 1)k ≡ 0 mod q 2 − 1. By Lemma 3.2(i), it follows that gcd(pt/2 + 1, (q +
1)(q − 1)) = gcd(pt/2 + 1, q + 1) = pt/2 + 1. So (q − 1) · p(q+1)
t/2 +1 |k. But then q − 1|k,
a contradiction to the definition of G I3 . Hence in this case U2 = ∅. So
 t
if 2t | n,

p − 2
t
|C G I3 (H)| = |U1 | + |U2 | = (p − 3)/2
if t | n and 2t n,

 t
t/2
(p − p − 2)/2 if t n.
Next we calculate |C G I22 (H)|. Let
{{(k, l), (l, k)} ∈ C G I22 (H) | pt k ≡ k, pt l ≡ l} if i = 1,
Ui :=
{{(k, l), (l, k)} ∈ C G I22 (H) | pt k ≡ l, pt l ≡ k} if i = 2.
If x = {(k, l), (l, k)} ∈ G I22 , then x ∈ U1 if and only if (pt −1)k ≡ 0 and (pt −1)l ≡ 0
mod q + 1.
Suppose t | n. By Lemma 3.2(i), this is equivalent with 2k ≡ 0 and 2l ≡ 0 mod
q + 1. By the definition of G I22 , we get |U1 | = 1.
Suppose t n. By Lemma 3.2(i), this is equivalent with k, l are multiples of
q+1
. So we get |U1 | = pt/2 (pt/2 + 1)/2.
pt/2 +1
Suppose t | n. If x = {(k, l), (l, k)} ∈ G I22 , then x ∈ U2 if and only if pt k ≡ l and
t
p l ≡ k mod q + 1. From these two congruences, we get (p2t − 1)k ≡ 0 mod q + 1. By
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Lemma 3.2(i), this is equivalent with 2(pt + 1)k ≡ 0 mod q + 1. By Lemmas 3.2(iii)
and 3.4, this equivalent with 2k ≡ 0 mod q + 1 if 2t | n, and (pt + 1)k ≡ 0 mod
q + 1 if 2t n. Thus, we have
0
if 2t | n,
|U2 | =
(pt − 1)/2 if 2t n.
Suppose t n. If x = {(k, l), (l, k)} ∈ G I22 , then x ∈ U2 implies (p2t − 1)k ≡ 0
and (p2t −1)l ≡ 0 mod q+1. By Lemma 3.2(iii), this is equivalent with 2(pt −1)k ≡ 0
mod q + 1. By Lemma 3.5, this is equivalent with (pt − 1)k ≡ 0 mod q + 1. Then
k ≡ pt k ≡ l mod q + 1, a contradiction to the definition of G I22 . Hence in this case
U2 = ∅. So

if 2t | n,

1
t
|C G I22 (H)| = |U1 | + |U2 | = (p + 1)/2
if t | n and 2t n,

 t/2 t/2
p (p + 1)/2 if t n.
So, in both cases, |CI (H)| = |C G I3 (H)| + |C G I22 (H)| = pt − 1.
Let I ∈ {G I7 ∪ G I16 , G I8 ∪ G I15 }. Then these unions of parameter sets are
isomorphic H-sets, so that we can assume I = G I7 ∪ G I16 . First, we compute
|C G I7 (H)|. Let
if i = 1,
{{(k, l), (−qk, l)} ∈ C G I7 (H) | pt k ≡ k, pt l ≡ l}
Ui :=
t
t
{{(k, l), (−qk, l)} ∈ C G I7 (H) | p k ≡ −qk, p l ≡ l} if i = 2.
If x = {(k, l), (−qk, l)} ∈ G I7 , then x ∈ U1 if and only if (pt − 1)k ≡ 0 mod q 2 − 1
and (pt − 1)l ≡ 0 mod q + 1.
2
−1
Suppose t | n. By Lemma 3.2(i) and (ii), this is equivalent with qpt −1
| k and
t
2l ≡ 0 mod q + 1. Hence, |U1 | = p − 3.
2
−1
Suppose t n. By Lemma 3.2(i) and (ii), this is equivalent with qpt −1
| k and
q+1
t
t/2
t/2
|
l.
So
by
the
definition
of
I
,
we
get
|U
|
=
(p
−
p
−
2)(p
+
1)/2.
G 7
1
pt/2 +1
Suppose t | n. If x = {(k, l), (−qk, l)} ∈ G I7 , then x ∈ U2 if and only if (pt +q)k ≡
0 mod q 2 −1 and (pt −1)l ≡ 0 mod q+1. Then (pn−t +1)k ≡ 0 mod q 2 −1 and 2l ≡ 0
mod q + 1. By Lemma 3.3, the first congruence is equivalent with (pt + 1)k ≡ 0
mod q 2 − 1 if 2t | n, and 2k ≡ 0 mod q 2 − 1 if 2t n. So by the definition of G I7 ,
we get
pt − 1 if 2t | n,
|U2 | =
0
if 2t n.
Suppose t n. If x = {(k, l), (−qk, l)} ∈ G I7 , then x ∈ U2 if and only if (pt +q)k ≡
0 mod q 2 −1 and (pt −1)l ≡ 0 mod q+1. Then the first congruence is equivalent with
(pn−t + 1)k ≡ 0 mod q 2 − 1. By Lemma 3.3, this is equivalent with (pt/2 + 1)k ≡ 0
mod q 2 −1. By Lemma 3.2(i), it follows that gcd(pt/2 +1, (q+1)(q−1)) = gcd(pt/2 +1,
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q + 1) = pt/2 + 1. So (q − 1) · p(q+1)
t/2 +1 | k. But then q − 1 | k,
definition of G I7 . Hence in this case U2 = ∅. So
 t

2(p − 2)
|C G I7 (H)| = |U1 | + |U2 | = pt − 3

 t
(p − pt/2 − 2)(pt/2 + 1)/2
a contradiction to the
if 2t | n,
if t | n and 2t n,
if t n.
Next we calculate |C G I16 (H)|. Let
{{(k, l, m), (k, m, l)} ∈ C G I16 (H) | pt k ≡ k, pt l ≡ l, pt m ≡ m} if i = 1,
Ui :=
{{(k, l, m), (k, m, l)} ∈ C G I16 (H) | pt k ≡ k, pt l ≡ m, pt m ≡ l} if i = 2.
If x = {(k, l, m), (k, m, l)} ∈ G I16 , then x ∈ U1 if and only if (pt − 1)k ≡ 0,
(pt − 1)l ≡ 0 and (pt − 1)m ≡ 0 mod q + 1.
Suppose t | n. By Lemma 3.2(i), this is equivalent with 2k ≡ 0, 2l ≡ 0 and
2m ≡ 0 mod q + 1. By the definition of G I16 , we get |U1 | = 0.
Suppose t n. By Lemma 3.2(i), this is equivalent with k, l and m are multiples
q+1
. So we get |U1 | = pt/2 (pt − 1)/2.
of pt/2
+1
Suppose t | n. If x = {(k, l, m), (k, m, l)} ∈ G I16 , then x ∈ U2 if and only if
t
(p − 1)k ≡ 0, pt l ≡ m and pt m ≡ l mod q + 1. From the last two congruences we
get (pt − 1)k ≡ 0 and (p2t − 1)l ≡ 0 mod q + 1. By Lemma 3.2(i), this is equivalent
with 2k ≡ 0 and 2(pt + 1)l ≡ 0 mod q + 1. By Lemmas 3.2(iii) and 3.4, the second
congruence is equivalent with 2l ≡ 0 mod q + 1 if 2t | n, and (pt + 1)l ≡ 0 mod q + 1
if 2t n. So by the definition of G I16 , we get
0
if 2t | n,
|U2 | =
t
p − 1 if 2t n.
Suppose t n. If x = {(k, l, m), (k, m, l)} ∈ G I16 , then x ∈ U2 if and only if
(pt − 1)k ≡ 0, pt l ≡ m and pt m ≡ l mod q + 1. From the last two congruences
we get (pt − 1)k ≡ 0 and (p2t − 1)l ≡ 0 mod q + 1. By Lemma 3.2(iii), the second
congruence is equivalent with 2(pt − 1)l ≡ 0 mod q + 1. By Lemma 3.5, this is
equivalent with (pt −1)l ≡ 0 mod q +1. Then l ≡ pt l ≡ m mod q +1, a contradiction
to the definition of G I16 . Hence in this case U2 = ∅. So

if 2t | n,

0
|C G I16 (H)| = |U1 | + |U2 | = pt − 1
if t | n and 2t n,

 t/2 t
p (p − 1)/2 if t n.
Thus,
|CI (H)| = |C G I7 (H)| + |C G I16 (H)| =
2(pt − 2)
(p
t/2
if t | n,
+ 1)(p − p
t
t/2
− 1) if t n.
Let I ∈ {G I17 , G I18 , G I19 , G I21 }. If (k, l) ∈ I, then (k, l) ∈ CI (H) if and only if
(pt − 1)k ≡ 0 and (pt − 1)l ≡ 0 mod q + 1.
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Suppose t | n. By Lemma 3.2(i), this is equivalent with 2k ≡ 0 and 2l ≡ 0 mod
q + 1. So by the definition of I, we get |CI (H)| = 2.
Suppose t n. By Lemma 3.2(i), this is equivalent with k, l are multiples of
q+1
. So we get |CI (H)| = pt/2 (pt/2 + 1).
pt/2 +1
Let I = B I1 . If (k, l) ∈ I, then (k, l) ∈ CI (H) if and only if (pt − 1)k ≡ 0 and
t
(p − 1)l ≡ 0 mod q 2 − 1. By Lemma 3.2(ii), this is equivalent with k, l are multiples
2
−1
of qpt −1
. Thus, |CI (H)| = (pt − 1)2 .
Let I ∈ {B I2 , P I4 , Q I1 , Q I2 }. If (k, l) ∈ I, then (k, l) ∈ CI (H) if and only if
(pt − 1)k ≡ 0 mod q 2 − 1 and (pt − 1)l ≡ 0 mod q + 1.
2
−1
| k and
Suppose t | n. By Lemma 3.2(i) and (ii), this is equivalent with qpt −1
2
−1
2l ≡ 0 mod q + 1. So we get CI (H) = {(k, l) ∈ I | qpt −1
| k and
|CI (H)| = 2(pt − 1).
Suppose t n. By Lemma 3.2(i) and (ii), this is equivalent with
(pt/2 + 1)l ≡ 0 mod q + 1. So we get CI (H) = {(k, l) ∈ I |
q2 −1
pt −1
|k
q+1
2
| l} and
q2 −1
pt −1 | k
q+1
and pt/2
+1
and
| l}
and |CI (H)| = (p − 1)(p + 1).
Let I ∈ {B I3 , B I6 , P I1 , P I2 , Q I4 }. If k ∈ I, then k ∈ CI (H) if and only if (pt −
1)k ≡ 0 mod q 2 − 1. So we get CI (H) = {k ∈ I | k is a multiple of (q 2 − 1)/(pt − 1)}
and |CI (H)| = pt − 1.
Let I = B I5 . If (k, l) ∈ I, then (k, l) ∈ CI (H) if and only if (pt − 1)k ≡ 0 mod
q + 1 and (pt − 1)l ≡ 0 mod q 2 − 1.
Suppose t | n. By Lemma 3.2(i) and (ii), this is equivalent with 2k ≡ 0 mod
2
2
−1
−1
| l. So we get CI (H) = {(k, l) ∈ I | q+1
| k and qpt −1
| l} and
q + 1 and qpt −1
2
t
|CI (H)| = 2(p − 1).
Suppose t n. By Lemma 3.2(i) and (ii), this is equivalent with (pt/2 + 1)k ≡ 0
2
2
−1
q+1
−1
mod q + 1 and qpt −1
| l. So we get CI (H) = {(k, l) ∈ I | pt/2
| k and qpt −1
| l} and
+1
t
t/2
|CI (H)| = (pt/2 + 1)(pt − 1).
Let I ∈ {B I7 , Q I7 , Q I9 , Q I10 }. If (k, l) ∈ I, then (k, l) ∈ CI (H) if and only if
(pt − 1)k ≡ 0 and (pt − 1)l ≡ 0 mod q + 1.
Suppose t | n. By Lemma 3.2(i), this is equivalent with 2k ≡ 0 and 2l ≡ 0 mod
q + 1. So we get CI (H) = {(k, l) ∈ I | q+1
2 | k, l} and |CI (H)| = 4.
Suppose t n. By Lemma 3.2(i), this is equivalent with k, l are multiples of
q+1
. So we get |CI (H)| = (pt/2 + 1)2 .
pt/2 +1
Let I = P I3 ∪ P I8 . First, we compute |C P I3 (H)|. Let
{{(k, l), (l, k)} ∈ C P I3 (H) | pt k ≡ k, pt l ≡ l} if i = 1,
Ui :=
{{(k, l), (l, k)} ∈ C P I3 (H) | pt k ≡ l, pt l ≡ k} if i = 2.
If x = {(k, l), (l, k)} ∈ P I3 , then x ∈ U1 if and only if (pt − 1)k ≡ 0 and (pt − 1)l ≡ 0
2
−1
mod q 2 − 1. By Lemma 3.2(ii), this is equivalent with k, l are multiples of qpt −1
.
Hence |U1 | = (pt − 2)(pt − 1)/2.
Suppose t | n. If x = {(k, l), (l, k)} ∈ P I3 , then x ∈ U2 if and only if pt k ≡ l and
t
p l ≡ k mod q 2 − 1. From these two congruences we get (p2t − 1)k ≡ 0 mod q 2 − 1,
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and this is equivalent to k being a multiple of (q 2 − 1)/(p2t − 1). Excluding those
solutions with k ≡ l mod q 2 − 1, we get |U2 | = pt (pt − 1)/2.
Suppose t n. If x = {(k, l), (l, k)} ∈ P I3 , then x ∈ U2 if and only if pt k ≡ l
and pt l ≡ k mod q 2 − 1. From these two congruences we get (p2t − 1)k ≡ 0 mod
q 2 − 1. By Lemma 3.2(iv), this is equivalent with 2(pt − 1)k ≡ 0 mod q 2 − 1. By
Lemma 3.6, this is equivalent with (pt − 1)k ≡ 0 mod q 2 − 1. Then k ≡ pt k ≡ l, a
contradiction to the definition of P I3 . Hence, U2 = ∅. So
(pt − 1)2
if t | n,
|C P I3 (H)| = |U1 | + |U2 | =
t
t
(p − 2)(p − 1)/2 if t n.
Next we calculate |C P I8 (H)|. If x = {k, q 2 k} ∈ P I8 , then x ∈ C P I8 (H) if and only
if (pt − 1)k ≡ 0 or (pt − q 2 )k ≡ 0 mod (q 2 + 1)(q 2 − 1). Suppose (pt − 1)k ≡ 0. By
Lemma 3.2(ii), it follows that gcd(pt −1, (q 2 +1)(q 2 −1)) = gcd(pt −1 , q 2 −1) = pt −1.
2
−1)
Thus (q 2 + 1) · (qpt −1
| k. But then (q 2 + 1) | k, a contradiction to the definition
of P I8 . So we have proved that x ∈ C P I8 (H) if and only if (pt − q 2 )k ≡ 0 mod
(q 2 + 1)(q 2 − 1).
Suppose t | n. If {k, q 2 k} ∈ C P I8 (H), then (pt − q 2 )k ≡ 0 mod (q 2 + 1)(q 2 − 1).
Thus (pt + 1)k ≡ 0 mod q 2 + 1 and (pt − 1)k ≡ 0 mod q 2 − 1. By Lemmas 3.2(ii)
2
2
2
2
−1
−1
and (v), we get q 2+1 | k and qpt −1
| k. Since q 2+1 | q 2 + 1 and qpt −1
| q 2 − 1 and since
pt − 1 is even, we have gcd( q
2
+1
2
2n
,
q2 −1
pt −1
q2 +1 q2 −1
2 · pt −1 | k. The condition
q2 −1
that pt −1 is even. Thus q 2 + 1 | k,
) = 1 and so
t | n implies (pt − 1)(pt + 1) | p − 1 = q 2 − 1, so
a contradiction to the definition of P I8 . Hence in this case C P I8 (H) = ∅.
Suppose t n. We claim
(q 2 + 1)(q 2 − 1)
2
C P I8 (H) = {k, q k} ∈ P I8 | k is a multiple of t
.
(p + 1)(pt − 1)
2
2
+1)(q −1)
Let k = (q
(pt +1)(pt −1) · m for some m ∈ Z. Because t | 2n and t n we have
2t | 2n − t. Since (pt − 1)(pt + 1) = p2t − 1 | p2n−t − 1 we then get (p2n−t − 1)k =
p2n−t −1
2
2
2
2
t
2
(pt +1)(pt −1) (q + 1)(q − 1) · m ≡ 0 mod (q + 1)(q − 1). So (p − q )k ≡ 0 mod
2
2
2
(q + 1)(q − 1) and {k, q k} ∈ C P I8 (H).
Conversely, suppose {k, q 2 k} ∈ C P I8 (H). Then (pt − q 2 )k ≡ 0 mod (q 2 + 1)(q 2 −
1). Hence (pt + 1)k ≡ 0 mod q 2 + 1 and (pt − 1)k ≡ 0 mod q 2 − 1. By Lemma 3.2(ii)
2
2
2
+1
−1
+1
and (v), this is equivalent with qpt +1
| k and qpt −1
| k. Since qpt +1
| q 2 + 1 and
q2 −1
pt −1
| q 2 − 1 and since
2
2
q2 −1
pt −1
2
2
+1 q −1
is odd by Lemma 3.6, we have gcd( qpt +1
, pt −1 ) = 1.
+1)(q −1)
Therefore (q
(pt +1)(pt −1) | k, and the claim holds. So by the definition of P I8 , we get
|C P I8 (H)| = pt (pt − 1)/2.
So in both cases, |CI (H)| = |C P I3 (H)| + |C P I8 (H)| = (pt − 1)2 .
Let I = Q I3 ∪ Q I8 . First, we compute |C Q I3 (H)|. Let
if i = 1,
{{(k, l), (k, −ql)} ∈ C Q I3 (H) | pt k ≡ k, pt l ≡ l}
Ui :=
t
t
{{(k, l), (k, −ql)} ∈ C Q I3 (H) | p k ≡ k, p l ≡ −ql} if i = 2.
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If x = {(k, l), (k, −ql)} ∈ Q I3 , then x ∈ U1 if and only if (pt − 1)k ≡ 0 and
(pt − 1)l ≡ 0 mod q 2 − 1. By Lemma 3.2(ii), this is equivalent with k, l being
2
−1
multiples of qpt −1
. So by the definition of Q I3 , we get
if t | n,
(pt − 1)(pt − 3)/2
|U1 | =
t
t
t/2
(p − 1)(p − p − 2)/2 if t n.
Suppose t | n. If x = {(k, l), (k, −ql)} ∈ Q I3 , then x ∈ U2 if and only if (pt −1)k ≡
2
−1
0 and (pt + q)l ≡ 0 mod q 2 − 1. Then qpt −1
| k and (pn−t + 1)l ≡ 0 mod q 2 − 1. By
Lemma 3.3, the second congruence is equivalent with (pt + 1)l ≡ 0 mod q 2 − 1 if
2t | n, and 2l ≡ 0 mod q 2 − 1 if 2t n. So by the definition of Q I3 , we get
(pt − 1)2 /2 if 2t | n,
|U2 | =
0
if 2t n.
Suppose t n. If x = {(k, l), (k, −ql)} ∈ Q I3 , then {(k, l), (k, −ql)} ∈ U2 if and only
2
−1
if (pt − 1)k ≡ 0 and (pt + q)l ≡ 0 mod q 2 − 1. Then qpt −1
| k and (pn−t + 1)l ≡ 0 mod
q 2 − 1. By Lemma 3.3, the second congruence is equivalent with (pt/2 + 1)l ≡ 0 mod
q 2 − 1. By Lemma 3.2(i), it follows that gcd(pt/2 + 1, (q + 1)(q − 1)) = gcd(pt/2 +
1, q + 1) = pt/2 + 1. So (q − 1) · p(q+1)
t/2 +1 | l. But then q − 1 | l, a contradiction to the
definition of Q I3 . Hence in this case U2 = ∅. So
 t
t
if 2t | n,

(p − 1)(p − 2)
t
t
|C Q I3 (H)| = |U1 | + |U2 | = (p − 1)(p − 3)/2
if t | n and 2t n,

 t
t
t/2
(p − 1)(p − p − 2)/2 if t n.
Next we calculate |C Q I8 (H)|. Let
{{(k, l, m), (k, m, l)} ∈ C Q I8 (H) | pt k ≡ k, pt l ≡ l, pt m ≡ m} if i = 1,
Ui :=
{{(k, l, m), (k, m, l)} ∈ C Q I8 (H) | pt k ≡ k, pt l ≡ m, pt m ≡ l} if i = 2.
If x = {(k, l, m), (k, m, l)} ∈ Q I8 , then x ∈ U1 if and only if (pt − 1)k ≡ 0 mod
q 2 − 1, and (pt − 1)l ≡ 0 and (pt − 1)m ≡ 0 mod q + 1.
Suppose t | n. Then (pt − 1)k ≡ 0 mod q 2 − 1, and 2l ≡ 0 and 2m ≡ 0 mod
2
−1
q + 1. This is equivalent with qpt −1
| k, and q+1
2 | l, m. So by the definition of Q I8 ,
t
we get |U1 | = p − 1.
Suppose t n. Then (pt −1)k ≡ 0 mod q 2 −1, and (pt/2 +1)l ≡ 0 and (pt/2 +1)m ≡
0 mod q + 1. Hence
q+1
q2 − 1
| l, m
| k and t/2
U1 = {(k, l, m), (k, m, l)} ∈ Q I8 | t
p −1
p +1
and |U1 | = pt/2 (pt/2 + 1)(pt − 1)/2.
Suppose t | n. If x = {(k, l, m), (k, m, l)} ∈ Q I8 , then x ∈ U2 if and only if
t
(p − 1)k ≡ 0 mod q 2 − 1, pt l ≡ m and pt m ≡ l mod q + 1. By Lemma 3.2(i)
2
−1
| k and (p2t − 1)l ≡ 0 mod q + 1. By
we get from the last two congruences, qpt −1
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Lemma 3.2(i), the second congruence is equivalent with 2(pt + 1)l ≡ 0 mod q + 1.
By Lemmas 3.2(iii) and 3.4, this is equivalent with 2l ≡ 0 mod q + 1 if 2t | n, and
(pt + 1)l ≡ 0 mod q + 1 if 2t n. So by the definition of Q I8 , we get
0
if 2t | n,
|U2 | =
t
2
(p − 1) /2 if 2t n.
Suppose t n. If x = {(k, l, m), (k, m, l)} ∈ Q I8 , then x ∈ U2 if and only if
(p − 1)k ≡ 0 mod q 2 − 1, and pt l ≡ m and pt m ≡ l mod q + 1. By Lemma 3.2(i)
2
−1
we get from the last two congruences, qpt −1
| k and (p2t − 1)l ≡ 0 mod q + 1.
By Lemma 3.2(iii), the second congruence is equivalent with 2(pt − 1)l ≡ 0 mod
q + 1. By Lemma 3.5, this is equivalent with (pt − 1)l ≡ 0 mod q + 1. Then
l ≡ pt l ≡ m mod q + 1, a contradiction to the definition of Q I8 . Hence in this case
U2 = ∅. So
 t
if 2t | n,

p − 1
t
t
|C Q I8 (H)| = |U1 | + |U2 | = (p − 1)(p + 1)/2
if t | n and 2t n,

 t/2 t/2
t
p (p + 1)(p − 1)/2 if t n.
t
So in both cases, |CI (H)| = |C Q I3 (H)| + |C Q I8 (H)| = (pt − 1)2 .
Let I = Q I6 ∪ Q I11 . First, we compute |C Q I6 (H)|. Let
if i = 1,
{{(k, l), (k, −ql)} ∈ C Q I6 (H) | pt k ≡ k, pt l ≡ l}
Ui :=
t
t
{{(k, l), (k, −ql)} ∈ C Q I6 (H) | p k ≡ k, p l ≡ −ql} if i = 2.
If x = {(k, l), (k, −ql)} ∈ Q I6 , then x ∈ U1 if and only if (pt − 1)k ≡ 0 mod q + 1
and (pt − 1)l ≡ 0 mod q 2 − 1.
Suppose t | n. By Lemma 3.2(i) and (ii), this is equivalent with q+1
2 | k and
q2 −1
pt −1
| l. So we get |U1 | = pt − 3.
Suppose t n. Then
q+1
pt/2 +1
| k and
q2 −1
pt −1
| l. So we get |U1 | = (pt/2 + 1)(pt −
pt/2 − 2)/2.
Suppose t | n. If x = {(k, l), (k, −ql)} ∈ Q I6 , then x ∈ U2 if and only if (pt −1)k ≡
n−t
0 mod q + 1 and (pt + q)l ≡ 0 mod q 2 − 1. Then q+1
+ 1)l ≡ 0 mod
2 | k and (p
2
t
q − 1. By Lemma 3.3, the second congruence is equivalent with (p + 1)l ≡ 0 mod
q 2 − 1 if 2t | n, and 2l ≡ 0 mod q 2 − 1 if 2t n. So by the definition of Q I6 ,
we get
pt − 1 if 2t | n,
|U2 | =
0
if 2t n.
Suppose t n. If {(k, l), (k, −ql)} ∈ U2 , then (pt −1)k ≡ 0 mod q+1 and (pt +q)l ≡ 0
q+1
| k and (pn−t + 1)l ≡ 0 mod q 2 − 1. By Lemma 3.3, the
mod q 2 − 1. Then pt/2
+1
second congruence is equivalent with (pt/2 + 1)l ≡ 0 mod q 2 − 1. By Lemma 3.2(i),
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it follows that gcd(pt/2 + 1, (q + 1)(q − 1)) = gcd(pt/2 + 1, q + 1) = pt/2 + 1. So
(q − 1) · p(q+1)
t/2 +1 | l. But then q − 1 | l, a contradiction to the definition of Q I6 . Hence
in this case U2 = ∅. So
 t
if 2t | n,

2(p − 2)
t
|C Q I6 (H)| = |U1 | + |U2 | = p − 3
if t | n and 2t n,

 t/2
t
t/2
(p + 1)(p − p − 2)/2 if t n.
Next we calculate |C Q I11 (H)|. Let
{{(k, l, m), (k, m, l)} ∈ C Q I11 (H) | pt k ≡ k, pt l ≡ l, pt m ≡ m} if i = 1,
Ui :=
{{(k, l, m), (k, m, l)} ∈ C Q I11 (H) | pt k ≡ k, pt l ≡ m, pt m ≡ l} if i = 2.
If x = {(k, l, m), (k, m, l)} ∈ Q I11 , then x ∈ U1 if and only if (pt − 1)k ≡ 0,
(pt − 1)l ≡ 0 and (pt − 1)m ≡ 0 mod q + 1.
Suppose t | n. Then 2k ≡ 0, 2l ≡ 0 and 2m ≡ 0 mod q + 1. So by the definition
of Q I11 , we get |U1 | = 2.
Suppose t n. Then (pt/2 + 1)k ≡ 0, (pt/2 + 1)l ≡ 0 and (pt/2 + 1)m ≡ 0 mod
q + 1. Hence
q+1
| k, l, m
U1 = {(k, l, m), (k, m, l)} ∈ Q I11 | t/2
p +1
and |U1 | = pt/2 (pt/2 + 1)2 /2.
Suppose t | n. If x = {(k, l, m), (k, m, l)} ∈ Q I11 , then x ∈ U2 if and only if
(pt − 1)k ≡ 0, pt l ≡ m and pt m ≡ l mod q + 1. By Lemma 3.2(i) and the last two
2t
congruences, we get q+1
2 | k and (p − 1)l ≡ 0 mod q + 1. By Lemma 3.2(i), the
second congruence is equivalent with 2(pt + 1)l ≡ 0 mod q + 1. By Lemmas 3.2(iii)
and 3.4, this is equivalent with 2l ≡ 0 mod q + 1 if 2t | n, and (pt + 1)l ≡ 0 mod
q + 1 if 2t n. So by the definition of Q I8 , we get
0
if 2t | n,
|U2 | =
t
p − 1 if 2t n.
Suppose t n. If x = {(k, l, m), (k, m, l)} ∈ Q I11 , then x ∈ U2 if and only if
(p − 1)k ≡ 0, pt l ≡ m and pt m ≡ l mod q + 1. By Lemma 3.2(i) and the last two
q+1
| k and (p2t − 1)l ≡ 0 mod q + 1. By Lemma 3.2 (iii),
congruences, we get pt/2
+1
the second congruence is equivalent with 2(pt − 1)l ≡ 0 mod q + 1. By Lemma 3.5,
this is equivalent with (pt − 1)l ≡ 0 mod q + 1. Then l ≡ pt l ≡ m mod q + 1, a
contradiction to the definition of Q I11 . Hence in this case U2 = ∅. So

if 2t | n,

2
t
|C Q I11 (H)| = |U1 | + |U2 | = p + 1
if t | n and 2t n,

 t/2 t/2
2
p (p + 1) /2 if t n.
t
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Thus,
|CI (H)| = |C Q I6 (H)| + |C Q I11 (H)| =
2(pt − 1)
(p
t/2
if t | n,
+ 1)(p − 1) if t n.
t
Now, we deal with the regular semisimple irreducible characters of G.
Proposition 4.2. Let G = GU4 (q 2 ), t | 2n, I := G I2 ∪ G I5 ∪ G I6 ∪ G I9 ∪ G I10 and
H = αt a subgroup of O. Then
(a) |CI (H)| is equal to the number of those regular semisimple conjugacy classes of
G which are stabilized by αt .
(b) If t n (respectively t | n), then |CI (H)| is equal to the number of regular
semisimple conjugacy classes of GU4 (pt ) (respectively Sp4 (pt )).
(c) |CI (H)| = (pt − 1)2 .
Proof. We use an argument similar to the one in [2, Proposition 4.2]. The set
I parameterizes the regular semisimple irreducible characters of G. We fix some
notation. Let Fq2 be a finite field with q 2 elements, F an algebraic closure of Fq2
and G = GL4 (F). Let α be the field automorphism α : G → G obtained from the
map F → F, x → xp , and γ : G → G the graph automorphism of order 2 given by
[13, p. 68]. Thus γ(g) = X(g T )−1 X −1 for any g ∈ G, where


0 0
0 1


 0 0 −1 0

.
X := 
0 0
 0 1

−1 0
0 0
F
F
Setting F := αn ◦ γ = γ ◦ αn we get G = GU4 (q 2 ) = G = GL4 (q 2 ) = {g ∈
G | F (g) = g}. Since the restriction α|G : G → G of α to G generates Out(G) we
can assume α|G = α.
For L ∈ {G, G}, let Sreg (L) be the set of all regular semisimple conjugacy classes
of L. If ρ is an endomorphism of L, then let Sreg (L)ρ := {C ∈ Sreg (L) | C ρ = C} be
the set of ρ-stable regular semisimple conjugacy classes of L. Finally, let Irrss
reg (G)
be the set of regular semisimple irreducible characters of G.
F
By [5, Corollary 3.10, p. 197] of Springer-Steinberg, the map C → C ∩ G is a
bijection from Sreg (G)F onto Sreg (G) and this bijection induces a bijection between
the set of regular semisimple conjugacy classes of G fixed by αt and the set of
F -stable regular semisimple conjugacy classes of G fixed by αt . It follows that,
since αt raises every element of a maximally split torus of G to its pt th power, the
automorphism αt maps each regular semisimple conjugacy class (g)G of G to the
t
class (g p )G . In other words, αt acts on the regular semisimple conjugacy classes
of G like the pt th power map (this does not mean, that αt maps every regular
semisimple element of G to its pt th power).
(a) Since G = GU4 (q 2 ) is isomorphic to its dual group (in the sense of [7,
t
Sec. 4.4, p. 120]), the number |Sreg (G)α | of fixed points of αt on Sreg (G) is
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equal to the number of fixed points of αt on Irrss
reg (G). By definition, the latter
equals |CI (H)|.
(b) In this part of the proof, we imitate an argument which is used in the proof of [4,
Lemma 4.1]. As we have seen at the beginning of this proof, there is a bijection from
t
the set of regular semisimple conjugacy classes of G fixed by αt onto Sreg (G)F,α ,
the set of fixed points of Sreg (G) under the action of the group F, αt . So by (a),
t
t
we have |CI (H)| = |Sreg (G)α | = |Sreg (G)F,α |.
Case 1. Suppose t n, then F, αt = αn ◦ γ, αt = αt/2 ◦ γ, αt = αt/2 ◦ γ.
t/2
αt/2 ◦γ
αt/2 ◦γ
∼
Thus |CI (H)| = |Sreg (G)α ◦γ | = |Sreg (G
)|. Since G
= GU4 (pt ), we get
|CI (H)| = |Sreg (GU4 (pt ))|, proving (b) in this case.
Case 2. Suppose t | n, then F, αt = αn ◦ γ, αt = γ, αt . Thus |CI (H)| =
t
t
|Sreg (G)F,α | = |(Sreg (G)α )γ | = |Sreg (G
(G
αt γ
) |. But G
αt
= GL4 (pt ), so
αt γ
) = GL4 (pt )γ = {g ∈ GL4 (pt ) | gXg T = X}.
Now X T = −X and so the form determined by X is symplectic. It follows that
(G
αt γ
) = Sp4 (pt ) and |CI (H)| = |Sreg (Sp4 (pt ))|.
(c) From the character tables of GU4 (pt ) and Sp4 (pt ) in [19] and [20] we get that if
t n (respectively t | n), then the number of regular semisimple conjugacy classes
of GU4 (pt ) (respectively Sp4 (pt )) is equal to (pt − 1)2 .
5. Dade’s Invariant Conjecture for GU4 (q 2 ), q Odd
In this section, we prove Dade’s invariant conjecture for G = GU4 (q 2 ) in the defining
characteristic p, where q = pn with an odd prime p. By [16, p. 152], G has only two
p-blocks, the principal block B0 = B0 (G) and one defect-0-block (consisting of the
Steinberg character). Hence we have to verify the conjecture for B0 . We will follow
the notation of Sec. 4.
By a corollary of the Borel–Tits theorem [6], the normalizers of radical psubgroups are parabolic subgroups. The radical p-chains of G (up to G-conjugacy)
are given in Table 1.
Table 1.
Radical p-chains of G.
C
NG (C)
NA (C)
A
C1
{1}
G
C2
{1} < Op (P )
P
P α
C3
{1} < Op (P ) < Op (B)
B
B α
C4
{1} < Op (Q)
Q
Q α
C5
{1} < Op (Q) < Op (B)
B
B α
C6
{1} < Op (B)
B
B α
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Since NG (C5 ) = NG (C6 ) and NA (C5 ) = NA (C6 ), it follows that for all d ∈ N
and u | 2n
k(NG (C5 ), B0 , d, u) = k(NG (C6 ), B0 , d, u).
Thus the contribution of C5 and C6 in the alternating sum of Dade’s invariant
conjecture is zero. So Dade’s invariant conjecture for G is equivalent to
k(G, B0 , d, u) + k(B, B0 , d, u) = k(P, B0 , d, u) + k(Q, B0 , d, u)
(5.1)
for all d ∈ N and u | 2n.
a p-block of G = GU4 (p2n ) with
Theorem 5.1. Let p > 2 be a prime and B
positive defect. Then B satisfies Dade’s invariant conjecture.
= B0 . Suppose u | 2n and
Proof. By the proceeding remarks, we can assume B
2n
t
set t := u and H := α . Let S ∈ {G, B, P, Q}. By the character tables in [19],
we have k(S, B0 , d, u) = 0 when d ∈ {3n, 4n, 5n, 6n}.
(i) If d = 3n, then by Tables A.2 and A.6, we have
4
|C G Ij (H)| =
k(G, B0 , d, u) =
(pt/2 + 1)2
j∈{13,18}
and
k(Q, B0 , d, u) = |C Q I9 (H)| =
if t n,
if t | n,
4
(p
if t | n,
t/2
2
+ 1)
if t n.
Thus (5.1) holds in this case.
(ii) If d = 4n, then Table A.3 implies, that (5.1) is equivalent to
|C G Ij (H)| +
|C B Ij (H)| =
|C P Ij (H)| +
|C Q Ij (H)|,
j∈JG
j∈JB
j∈JP
j∈JQ
(5.2)
where JG := {4, 12, 20}, JB := {5, 7}, JP := {2, 10} and JQ := {6, 10, 11}.
By Table A.6, the sums on both sides of Eq. (5.2) are equal to 3(pt + 1) or
pt/2 (pt/2 + 1)(pt/2 + 2) according as t | n or t n. Thus (5.1) also holds in this
case.
(iii) If d = 5n, then Table A.4 implies, that (5.1) is equivalent to (5.2) with JG :=
{8, 14, 15, 17, 21}, JB := {6, 8}, JP := {6, 7, 9} and JQ := {2, 7}. By Table A.6,
the sums on both sides of the Eq. (5.2) are also equal to 3(pt + 1) or pt/2 (pt/2 +
1)(pt/2 + 2) according as t | n or t n. Thus (5.1) holds in this case.
(iv) If d = 6n, then Table A.5 implies, that (5.1) is equivalent to (5.2) with JG :=
{1, 2, 3, 5, 6, 7, 9, 10, 16, 19, 22}, JB := {1, 2, 3, 4}, JP := {1, 3, 4, 5, 8} and JQ :=
{1, 3, 4, 5, 8}. By Table A.6, the sums on both sides of Eq. (5.2) are equal to
2pt (pt + 1) or 2p3t/2 (pt/2 + 1) according as t | n or t n. Thus (5.1) also holds
in this case.
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Appendix
Table A.1. Parameter sets for the irreducible characters of the parabolic subgroups
G = GU4 (q 2 ), B, P, Q.
Parameter
Set
Characters
Parameters
G I1
χ1 (k)
k = 0, . . . , q
G I2
χ2 (k, l)
G I3
= G I4
χ3 (k), χ4 (k)
Equivalence
relation
q+1
see the remarks in Sec. 4
k = 0, . . . , q 2 − 2
q − 1 k,
k = 0,
Number of
Characters
{k ≡ −qk}
(q 2 −q−2)(q 2 −q−4)
8
q 2 −q−2
2
q 2 −1
2
G I5
χ5 (k)
see the remarks in Sec. 4
q 2 (q 2 −1)
4
G I6
χ6 (k, l, m)
see the remarks in Sec. 4
q(q+1)(q 2 −q−2)
4
G I7
= G I8
χ7 (k, l),
χ8 (k, l)
k = 0, . . . , q 2 − 2
l = 0, . . . , q
q − 1 k,
k = 0,
{(k, l)
≡ (−qk, l)}
(q+1)(q 2 −q−2)
2
q 2 −1
2
G I9
χ9 (k, l)
see the remarks in Sec. 4
q(q+1)(q 2 −1)
3
G I10
χ10 (k, l, m, n)
see the remarks in Sec. 4
q(q 2 −1)(q−2)
24
G I11
= ···
= G I14
χ11 (k), . . . ,
χ14 (k)
k = 0, . . . , q
G I15
=
χ15 (k, l, m),
χ16 (k, l, m)
k, l, m = 0, . . . , q
k = l, m ; l = k, m;
m = k, l
G I17
= G I18
= G I19
χ17 (k, l),
χ18 (k, l),
χ19 (k, l)
k, l = 0, . . . , q
k = l
G I20
χ20 (k, l)
k, l = 0, . . . , q
k = l
G I21
χ21 (k, l)
k, l = 0, . . . , q
k = l
G I22
χ22 (k, l)
k, l = 0, . . . , q
k = l
B I1
B α1 (k, l)
k, l = 0, . . . , q 2 − 2
B I2
B α2 (k, l)
k = 0, . . . , q 2 − 2
l = 0, . . . , q
(q + 1)(q 2 − 1)
B I3
B α3 (k)
k = 0, . . . , q 2 − 2
q2 − 1
B I4
B α4 (k)
k = 0, . . . , q
q+1
B I5
B α5 (k, l)
k = 0, . . . , q
l = 0, . . . , q 2 − 2
(q + 1)(q 2 − 1)
B I6
B α6 (k)
k = 0, . . . , q 2 − 2
q2 − 1
G I16
q+1
{(k, l, m)
≡ (k, m, l)}
q(q 2 −1)
2
q(q + 1)
{(k, l) ≡ (l, k)}
q(q+1)
2
q(q + 1)
{(k, l) ≡ (l, k)}
q(q+1)
2
(q 2 − 1)2
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Table A.1.
(Continued )
Parameter
Set
Characters
Parameters
B I7
B α7 (k)
k, l = 0, . . . , q
(q + 1)2
B I8
B α8 (k)
k = 0, . . . , q
q+1
P β1 (k),
P β2 (k)
k = 0, . . . , q 2 − 2
q2 − 1
P I3
P β3 (k, l)
k, l = 0, . . . , q 2 − 2
k = l
P I4
P β4 (k, l)
k = 0, . . . , q 2 − 2
l = 0, . . . , q
(q + 1)(q 2 − 1)
P I5
P β5 (k)
k = 0, . . . , q
q+1
P I1 = P I2
P I6
P β6 (k)
0, . . . , q 2
k=
q − 1 k,
k = 0,
Equivalence
relation
−2
{(k, l) ≡ (l, k)}
Number of
Characters
(q 2 −1)(q 2 −2)
2
{k ≡ −qk}
q 2 −q−2
2
q 2 −1
2
P I7
P β7 (k, l)
k, l = 0, . . . , q
k = l
{(k, l) ≡ (l, k)}
q(q+1)
2
P I8
P β8 (k)
k = 0, . . . , q 4 − 2
q2 + 1 k
{k ≡ q 2 k}
q 2 (q 2 −1)
2
P I10
P β9 (k),
k = 0, . . . , q
q+1
Q I2
Q γ1 (k, l),
k = 0, . . . , q 2 − 2
l = 0, . . . , q
(q + 1)(q 2 − 1)
Q γ3 (k, l)
k, l = 0, . . . , q 2 − 2
q − 1 l,
P I9
=
Q I1
=
Q I3
P β10 (k)
Q γ2 (k, l)
l = 0,
{(k, l) ≡ (k, −ql)}
(q 2 −1)(q 2 −q−2)
2
q 2 −1
2
Q I4
Q γ4 (k)
k = 0, . . . , q 2 − 2
q2 − 1
Q I5
Q γ5 (k)
k = 0, . . . , q
q+1
Q I6
Q γ6 (k, l)
k = 0, . . . , q
l = 0, . . . , q 2 − 2
q−1l
l = 0,
Q γ7 (k, l)
k, l = 0, . . . , q
Q I8
Q γ8 (k, l, m)
k = 0, . . . , q 2 − 2
l, m = 0, . . . , q
l = m
Q γ9 (k, l),
k, l = 0, . . . , q
Q γ11 (k, l, m)
k, l, m = 0, . . . , q
l = m
=
Q I11
Q I10
Q γ10 (k, l)
(q+1)(q 2 −q−2)
2
q 2 −1
2
Q I7
Q I9
{(k, l) ≡ (k, −ql)}
(q + 1)2
{(k, l, m) ≡ (k, m, l)}
q(q+1)(q 2 −1)
2
(q + 1)2
{(k, l, m) ≡ (k, m, l)}
q(q+1)2
2
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Table A.2.
The irreducible characters of the chain normalizers in GU4 (q 2 ) of defect 3n.
Character
G
Q
Degree
P
Q
G I13
I1
q+1
A1
χ18 (k, l)
G I18
I2
q(q + 1)
A7
Q γ9 (k, l)
q 3 (q
−
P
Q
1)(q 2
+ 1)
− 1)
Q I9
I1 × I1
(q +
1)2
A11
The irreducible characters of the chain normalizers in GU4 (q 2 ) of defect 4n.
Degree
Param.
set
Param.
Nozawa
G I4
J1
Number
q 2 −q−2
2
Class
χ4 (k)
q 2 (q + 1)(q 3 + 1)
χ12 (k)
q 2 (q 2 + 1)
G I12
I1
q+1
A1
χ20 (k, l)
q 2 (q 2 + 1)(q 2 − q + 1)
G I20
I2
q(q+1)
2
A10
B α5 (k, l)
q 2 (q − 1)
B I5
I1 × J0
(q + 1)(q 2 − 1)
B3
B α7 (k, l)
q 2 (q − 1)2
B I7
I1 × I1
(q + 1)2
A9
P β2 (k)
q2
C1
P β10 (k)
q 2 (q
Q γ6 (k, l)
q 2 (q 2
Q γ10 (k, l)
q 2 (q − 1)
Q γ11 (k, l, m)
q 2 (q
−
1)(q 2
+ 1)
− 1)
−
1)2
P I2
J0
q2
P I10
I1
q+1
−1
Q I6
I1 × J1
Q I10
I1 × I1
(q+1)(q 2 −q−2)
2
(q + 1)2
I1 × I2
q(q+1)2
2
Q I11
C1
A1
B1
A11
A15
The irreducible characters of the chain normalizers in GU4 (q 2 ) of defect 5n.
Character
B
Class
q 3 (q
Table A.4.
G
Number
q 3 (q 2 − q + 1)
Character
B
Parameter
Nozawa
χ13 (k)
Table A.3.
G
Parameter
set
Degree
Param.
set
Param.
Nozawa
G I8
J2
Number
(q+1)(q 2 −q−2)
2
Class
χ8 (k, l)
q(q 2 + 1)(q 3 + 1)
χ14 (k)
q(q 2 − q + 1)
G I14
I1
q+1
χ15 (k, l, m)
q(q − 1)(q 2 + 1)(q 2 − q + 1)
G I15
I3
χ17 (k, l)
q(q − 1)2 (q 2 + 1)
G I17
I2
q(q 2 −1)
2
q(q + 1)
A8
χ21 (k, l)
q(q 2 + 1)(q 2 − q + 1)
G I21
I2
q(q + 1)
A8
B α6 (k)
q(q 2 − 1)
B I6
J0
(q + 1)2
C2
B I8
I1
q+1
A1
P I6
J1
1)(q 2
B α8 (k)
q(q −
P β6 (k)
q(q 4 − 1)
− 1)
B2
A1
A12
P β7 (k, l)
q(q −
P I7
I2
q 2 −q−2
2
q(q+1)
2
P β9 (k)
q(q − 1)(q 2 + 1)
P I9
I1
q+1
A1
Q γ2 (k, l)
q
Q I2
J0 × I1
(q + 1)(q 2 − 1)
B2
Q I7
I2
(q + 1)2
A8
Q γ7 (k, l)
q(q −
1)2 (q 2
1)(q 2
+ 1)
− 1)
C2
A7
377
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Table A.5. The irreducible characters of the chain normalizers in GU4 (q 2 ) of defect 6n. We
use the abbreviation η := (q − 1)(q 2 + 1)(q 2 − q + 1).
Character
G
χ1 (k)
χ2 (k, l)
Q
(q + 1)(q 2 + 1)(q 3 + 1)
(q +
(q +
1)(q 2
χ6 (k, l, m)
(q −
1)(q 2
χ7 (k, l)
(q 2
χ9 (k, l)
(q 2
χ10 (k, l, m, n)
(q − 1)η
χ16 (k, l, m)
χ19 (k, l)
χ5 (k)
P
1
1)(q 3
χ3 (k)
B
Degree
+ 1)
Param.
set
Param.
Nozawa
G I1
I1
G I2
J4
G I3
J1
−
1)(q 3
+ 1)
G I5
S1
+
1)(q 3
+ 1)
G I6
J3
Number
q+1
(q 2 −q−2)(q 2 −q−4)
8
q 2 −q−2
2
q 2 (q 2 −1)
4
q(q+1)(q 2 −q−2)
4
(q+1)(q 2 −q−2)
2
q(q+1)(q 2 −1)
3
q(q 2 −1)(q−2)
24
q(q 2 −1)
2
Class
A1
C3
C1
E1
B3
+
1)(q 3
+ 1)
G I7
J2
−
1)(q 4
− 1)
G I9
R2
G I10
I4
η
G I16
I3
(q − 1)(q 2 + 1)
G I19
I2
q(q + 1)
A8
χ22 (k, l)
(q 2 + 1)(q 2 − q + 1)
G I22
I2
q(q+1)
2
A11
B α1 (k, l)
1
B I1
J0 × J0
(q 2 − 1)2
C5
B α2 (k, l)
q−1
B I2
J0 × I1
(q + 1)(q 2 − 1)
B2
B α3 (k)
q2 − 1
B I3
J0
q2 − 1
C1
B I4
I1
q+1
A1
P I1
J0
q2 − 1
C1
(q 2 −1)(q 2 −2)
2
(q + 1)(q 2 − 1)
B1
B α4 (k)
(q −
P β1 (k)
1
1)(q 2
− 1)
C1
D1
A14
A12
P β3 (k, l)
q2 + 1
P I3
J0 × J0
P β4 (k, l)
(q − 1)(q 2 + 1)
P I4
J0 × I1
P β5 (k)
(q − 1)(q 4 − 1)
P I5
I1
q+1
A1
P β8 (k)
q2 − 1
P I8
S1
q 2 (q 2 −1)
2
E1
Q γ1 (k, l)
1
Q I1
J0 × I1
(q + 1)(q 2 − 1)
B2
C1
C5
Q γ3 (k, l)
q+1
Q I3
J0 × J1
Q γ4 (k)
(q + 1)(q 2 − 1)
Q I4
J0
(q 2 −1)(q 2 −q−2)
2
q2 − 1
Q γ5 (k)
(q 2 − 1)2
Q I5
I1
q+1
A1
q(q+1)(q 2 −1)
2
B5
Q γ8 (k, l, m)
q−1
Q I8
J0 × I2
C3
Table A.6. Number of fixed points of H = αt on parameter sets of the irreducible
characters of the parabolic subgroups of GU4 (q 2 ). The unions of parameter sets in
this table are disjoint unions.
Parameter set I
G I1
G I2
∪ G I5 ∪ G I6 ∪ G I9 ∪ G I10
G I3 ∪ G I22
G I4 ∪ G I20
G I7 ∪ G I16
G I8 ∪ G I15
Number of fixed points |CI (H)|
if t | n
if t n
2
(pt − 1)2
pt − 1
pt − 1
2(pt − 2)
2(pt − 2)
pt/2 + 1
(pt − 1)2
pt − 1
pt − 1
t/2
(p
+ 1)(pt − pt/2 − 1)
(pt/2 + 1)(pt − pt/2 − 1)
June 9, 2010 15:57 WSPC/S0218-1967
132-IJAC
S0218196710005546
Dade’s Invariant Conjecture for GU4 (q 2 )
Table A.6.
Parameter set I
G I12
G I13
G I14
G I17
G I18
G I19
G I21
B I1
B I2
B I3
B I4
B I5
B I6
B I7
B I8
P I1
P I2
P I3 ∪ P I8
P I4
P I5
P I6 ∪ P I7
P I9
P I10
Q I1
Q I2
Q I3 ∪ Q I8
Q I4
Q I5
Q I6 ∪ Q I11
Q I7
Q I9
Q I10
379
(Continued )
Number of fixed points |CI (H)|
if t | n
if t n
2
2
2
2
2
2
2
(pt − 1)2
2(pt − 1)
pt − 1
2
2(pt − 1)
pt − 1
4
2
pt − 1
pt − 1
(pt − 1)2
2(pt − 1)
2
pt − 1
2
2
2(pt − 1)
2(pt − 1)
(pt − 1)2
pt − 1
2
2(pt − 1)
4
4
4
pt/2 + 1
pt/2 + 1
pt/2 + 1
t/2
p (pt/2 + 1)
pt/2 (pt/2 + 1)
pt/2 (pt/2 + 1)
pt/2 (pt/2 + 1)
(pt − 1)2
(pt − 1)(pt/2 + 1)
pt − 1
pt/2 + 1
(pt/2 + 1)(pt − 1)
pt − 1
(pt/2 + 1)2
pt/2 + 1
pt − 1
pt − 1
(pt − 1)2
(pt − 1)(pt/2 + 1)
pt/2 + 1
pt − 1
pt/2 + 1
pt/2 + 1
t
(p − 1)(pt/2 + 1)
(pt − 1)(pt/2 + 1)
(pt − 1)2
pt − 1
pt/2 + 1
(pt/2 + 1)(pt − 1)
(pt/2 + 1)2
(pt/2 + 1)2
(pt/2 + 1)2
Acknowledgments
Part of this work was done while the second author visited Chiba University in
Japan. He wishes to express his sincere thanks to Professor Shigeo Koshitani for
his support and great hospitality. He also acknowledges the support of a JSPS
postdoctoral fellowship from the Japan Society for the Promotion of Science.
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