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Journal of Algebra and Its Applications
Vol. 16, No. 5 (2017) 1750111 (12 pages)
c World Scientific Publishing Company
DOI: 10.1142/S0219498817501110
Finite p-groups with the least number
of outer p-automorphisms
Alireza Abdollahi
Department of Mathematics, University of Isfahan
81746-73441 Isfahan, Iran
School of Mathematics
Institute for Research in Fundamental Sciences (IPM)
P. O. Box 19395-5746, Tehran, Iran
[email protected]
Marzieh Ahmadi
Department of Mathematics, University of Isfahan
81746-73441 Isfahan, Iran
m [email protected]
S. Mohsen Ghoraishi
Department of Mathematics
Shahid Chamran University of Ahvaz
Ahvaz, Iran
[email protected]
Received 11 February 2016
Accepted 2 June 2016
Published 12 July 2016
Communicated by S. Sidki
In this paper we study finite p-groups G for which the p-part of |Aut(G) : Inn(G)| has
the least possible value p. We characterize the groups in some special cases, including
p-groups of nilpotency class 2, of maximal class, of order at most p5 , with cyclic Frattini
subgroup and p-groups G in which |G : Z(G)| ≤ p4 .
Keywords: Finite p-group; outer automorphism; p-automorphism.
Mathematics Subject Classification: 20D45, 20D15
1. Introduction
Let G be a finite p-group of order greater than p. One of the fundamental properties
of finite p-groups, proved by Gaschütz [11], is that G admits a noninner automorphism of p-power order. This means that |Aut(G) : Inn(G)|p ≥ p. (For an integer
n, np denotes its p-part, that is, np = pk , where n = pk m and gcd(m, p) = 1.) So
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the problem of studying finite p-groups G in which the latter attains the minimum
value arises.
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Problem 1.1 ([3, Problem 1004]). Study the p-groups G such that |Aut(G) :
Inn(G)|p = p. (Old problem.)
This paper is devoted to studying some classes of finite p-groups satisfying
|Aut(G) : Inn(G)|p = p. The main results are summarized as follows.
Theorem 1.1 (Main Results). Let G be a finite p-group such that |Aut(G) :
Inn(G)|p = p. Then
(1) G is abelian if and only if G ∼
= Zp × Zp or G ∼
= Zp2 (see Theorem 3.1).
(2) G is of nilpotency class 2 if and only if |G| = p3 (see Theorem 3.2).
(3) G is of maximal class if and only if either G is of order p3 or G is isomorphic
to one of the following groups of order p4 :
(a) x, y, z | x9 = y 3 = z 3 = 1, [x, y] = x3 , [x, z] = y, [y, z] = 1,
2
(b) x, y, z | xp = y p = [y, z] = 1, xp = z p , [x, y] = xp , [x, z] = y, where p > 2,
2
(c) x, y, z | xp = y p = [y, z] = 1, [x, y] = xp , [x, z] = y, xαp = z p , where p > 2
and α is a quadratic nonresidue for p,
(d) x, y | x8 = y 2 = 1, [x, y] = x2 , the semidihedral group of order 16 (see
Theorem 3.3).
(4) G has cyclic Frattini subgroup if and only if either G is the semidihedral group
of order 16, or G is of order p3 (see Theorem 3.4).
(5) AutΦ (G) = Inn(G)CAutΦ (G) (Z(Φ(G)), if G has noncyclic Frattini subgroup (see
Corollary 3.3).
2
3
(6) G is of order p5 if and only if p > 2 and G = a, b | ap = 1, bp = 1, [b, a] = bp ,
(see Theorem 3.5).
(7) |G : Z(G)| ≤ p4 if and only if G is one of the groups of order ≤ p5 in items (3)
and (6) (see Corollary 3.4).
2. Preliminaries
Our notation are standard and can be found in [13]. We use the following well-known
facts in the proofs.
Fact 2.1 ([2, Lemma 2.1]). Let G be a finite group and N be a normal subgroup
of G such that G/N is abelian. Let G/N = x1 N × · · · × xd N , where x1 , . . . , xd ∈
G and d = d(G/N ). If u1 , . . . , ud ∈ Z(N ) such that
(xi ui )ni = xni i
1 ≤ i ≤ d,
(2.1)
[xi , uj ] = [xj , ui ] 1 ≤ i < j ≤ d,
where ni = o(xi N ), then the mapping xi → xi ui , 1 ≤ i ≤ d, can be extended to an
automorphism of G leaving N elementwise fixed.
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It is easy to see that the constructed automorphism in Fact 2.1 has order
lcm1≤i≤d o(ui ). Moreover, we remark that Fact 2.1 is well known in the case when
G/N is cyclic. In this case the condition (2.1) reduces to (x1 u1 )n1 = xn1 1 .
Fact 2.2 ([7, Remark 4]). Let X be a central product of subgroups A, B (i.e.
X = AB and [A, B] = 1). Suppose that α ∈ Aut(A) and β ∈ Aut(B) agree on
A ∩ B. Then α and β admit a common extension γ ∈ Aut(X). In particular, if
A has a noninner p-automorphism which fixes Z(A) elementwise, then X has a
noninner p-automorphism of the same order which fixes Z(A) and B elementwise.
The following is a generalization of Gaschütz’s theorem due to Schmid [18] (see
[1] for some related results).
Fact 2.3. If G is a nonabelian p-group, then G has a noninner p-automorphism
that fixes the center Z(G) elementwise.
Let IA(G) be the subgroup of Aut(G) consisting of all automorphisms of G
inducing the identity on G/G . Then IA(G) is a normal subgroup of Aut(G).
Fact 2.4 ([5, Theorem 3.2]). Let G = a, b be a metabelian, two-generated
group. Then the following conditions are equivalent:
(1) For all u, v ∈ G , there is an automorphism of G that maps a to au and b to bv;
(2) G is nilpotent.
By a theorem of Burnside [4, Theorem, p. 241], a nonabelian group whose center
is cyclic cannot be the derived group of a p-group. Thus the following fact is obvious.
Fact 2.5. All groups of order p5 are metabelian.
The following fact can be proved easily by induction.
Fact 2.6. Let G be a group of nilpotency class 3. Then for every positive integer
n and x, y ∈ G,
(1)
(2)
(3)
(4)
n
yxn = xn y[y, x]n [y, x, x]( 2 ) ,
n
y n x = xy n [y, x]n [y, x, y]( 2 ) ,
n
[y, xn ] = [y, x]n [y, x, x]( 2 ) ,
n
n
moreover, if [y, x, y] = 1 then (xy)n = xn y n [y, x]( 2 ) [y, x, x]( 3 ) .
Fact 2.7 ([15, Theorem 3.4]). Let G be a finite nonabelian p-group of maximal
class. Then |G| = |Aut(G)|p if and only if either G is of order p3 or G is isomorphic
to one of the following groups of order p4 :
(a) x, y, z | x9 = y 3 = z 3 = 1, [x, y] = x3 , [x, z] = y, [y, z] = 1,
2
(b) x, y, z | xp = y p = [y, z] = 1, xp = z p , [x, y] = xp , [x, z] = y, where p > 2,
2
(c) x, y, z | xp = y p = [y, z] = 1, [x, y] = xp , [x, z] = y, xαp = z p , where p > 2 and
α is a quadratic nonresidue for p,
(d) x, y | x8 = y 2 = 1, [x, y] = x2 , the semidihedral group of order 16.
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Fact 2.8 ([14, Theorem 4.1]). Let G be an abelian group of type (pe1 , pe2 , . . . ,
pen ), where 1 ≤ e1 ≤ e2 ≤ · · · ≤ en . Define the following 2n numbers: dk = max{l :
el = ek }, ck = min{l : el = ek }. Then
|Aut(G)| =
n
(pdk − pk−1 )
n
(pej )n−dj
j=1
k=1
n
(pei −1 )n−ci +1 .
i=1
Fact 2.9 ([10, Theorem 1.1]). Let G be a finite nonabelian p-group with cyclic
Frattini subgroup. Then |G| = |Aut(G)|p if and only if either G ∼
= S16 or Z(G) is
2
cyclic and |G/Z(G)| = p . In particular, if p = 2 then G has one of the following
types: S16 , D8 , Q8 , M2n , or L2n+2 , where
n
n−1
L2n+2 = x, y, z | x2 = (xy)2 = 1, z 2 = 1, [x, z] = [y, z] = 1, z 2
= y 2 ,
(n > 1).
Fact 2.10 ([19, Theorem 1.1]). If G is a finite p-group, then CAutΦ (G)(Z(Φ(G))) ≤
Inn(G) if and only if G is elementary abelian or Φ(G) = Z(G) and Z(G) is cyclic.
3. Proof of the Main Results
Let AutΦ (G) and AutΦ (G) denote the groups of automorphisms of G that fix
G/Φ(G) and Φ(G), respectively. Obviously, AutΦ (G) is a normal subgroup of
Aut(G) which contains Inn(G). By a well-known result of Burnside, AutΦ (G) is
a p-group whenever G is a finite p-group. Muller [16, Theorem] has shown that
if G is neither elementary abelian nor extraspecial, then AutΦ (G) properly contains Inn(G). Winter [22, Theorem 1] proved that AutΦ (G) is a normal Sylow
p-subgroup of Aut(G), when G is an extraspecial p-group. Therefore we have the
following remark.
Remark 3.1. Let G be a finite nonabelian p-group such that |Aut(G) : Inn(G)|p =
p. Then the outer automorphisms group of G, Aut(G)/Inn(G), has a normal Sylow
p-subgroup of order p.
We first consider the case of finite abelian p-groups.
Theorem 3.1. Let G be a finite abelian p-group. Then |Aut(G) : Inn(G)|p = p if
and only if G ∼
= Zp2 .
= Zp × Zp or G ∼
Proof. Let G be an abelian group. First suppose that |Aut(G) : Inn(G)|p = p.
Thus |Aut(G)| = p. Using the notation and result of Fact 2.8, we have
n
p( 2 ) =
n
pk−1 | |Aut(G)|.
k=1
Therefore, either n = 1 or n = 2. Another application of Fact 2.8 shows that in the
former case e1 = 2 and in the latter e1 = 1 = e2 . The converse is obvious.
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We remark that the automorphisms group of an elementary abelian group of
order p2 is isomorphic to GL(2, p) which has p + 1 Sylow p-subgroup.
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Lemma 3.1. Let G be a finite nonabelian p-group. If G is a central product of
nontrivial proper subgroups A and B, then |Aut(G) : Inn(G)|p > p.
Proof. We may assume that A is nonabelian. Then by Fact 2.3, A has a noninner
p-automorphism α that fixes the center Z(A) elementwise. By Fact 2.2, α extends
to a noninner p-automorphism γ of G which fixes both Z(A) and B elementwise.
We may also assume that αp ∈ Inn(A). Hence γ p ∈ Inn(G).
Now we consider two cases. First, suppose that B is nonabelian. Similar to the
latter argument, G has a noninner p-automorphism δ which fixes both Z(B) and A
elementwise and δ p ∈ Inn(G). Thus in this case we have [γ, δ] = 1 and γ ∩ δ = 1.
Next, suppose that B is abelian. Let M be a maximal subgroup of G which
contains A. Also let g ∈ B\M and z ∈ Z(A) be an element of order p. Since
M = A(B ∩ M ), we have Z(M ) = Z(A)(B ∩ M ). Thus z ∈ Z(M ) and it follows
from Fact 2.1 that the map given by g → gz, can be extended to an automorphism,
δ, of G of order p that fixes M elementwise. It is easy to check that δ is noninner.
Now let x ∈ G. Then x = g i m for some integer 0 ≤ i ≤ p − 1 and m ∈ M . Since
M γ = M , we have
xγδ = ((g i m)γ )δ = (g i mγ )δ = (g i )δ mγ = (gz)i mγ = ((gz)i m)γ = xδγ .
Thus [γ, δ] = 1, γ ∩ δ = 1.
Therefore in the both cases γ, δInn(G)/Inn(G) is elementary abelian of order
p2 and the result follows.
The following corollary is now straightforward.
Corollary 3.1. Let G be a finite nonabelian p-group. If Z(G) ≤ Φ(G), then
|Aut(G) : Inn(G)|p > p.
Lemma 3.2. Let G be a finite p-group such that |G| | |Aut(G)|. Then |Aut(G) :
Inn(G)|p = p if and only if Z(G) is of order p and |Aut(G)|p = |G|.
Proof. If |G| | |Aut(G)|, then
|Z(G)| = |G|/|Inn(G)| ≤ |Aut(G) : Inn(G)|p .
This implies the result.
Remark 3.2. If G is a p-group of maximal class, then Z(G) is of order p and it follows form the result of Gaschütz, mentioned in the first section, that |G| | |Aut(G)|.
The latter divisibility condition holds for various classes of finite p-groups, including
p-groups of nilpotency class 2 [9], p-groups with cyclic Frattini subgroup [8, Corollary 1.1], p-groups G in which |G : Z(G)| ≤ p4 [6, Theorem], etc. It was conjectured
that the divisibility condition holds for every finite p-group of order ≥ p3 . This
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was known as the divisibility conjecture. However, recently, it turns out that the
conjecture does not hold in general. For more details on the subject see [12].
According to a well-known theorem of Hall [17, Theorem 5.3.3], if G is a
finite p-group of order pn and d is the minimum number of generators of G, then
|Aut(G)|p ≤ pm , where m = d(2n − d − 1)/2. If we apply this with n = 3 and d = 2,
we get |Aut(G)|p ≤ p3 for a nonabelian group G of order p3 . As a consequence we
have the following fact.
Fact 3.1. Let G be a nonabelian group of order p3 . Then |Aut(G) : Inn(G)|p = p.
We remark that one can conclude Fact 3.1 from [22, Theorem 1].
Theorem 3.2. Let G be a finite nonabelian p-group of class 2. Then |Aut(G) :
Inn(G)|p = p if and only if |G| = p3 .
Proof. Suppose that |Aut(G) : Inn(G)|p = p. By Remark 3.2, we know that
|G| | |Aut(G)|. Hence it follows from Lemma 3.2 that Z(G) has order p. Since G
is nilpotent of class 2, we have G ≤ Z(G) and exp(G/Z(G)) = exp(G ). Thus G
is an extraspecial p-group and therefore it is a central product of nonabelian subgroups of order p3 . Now Lemma 3.1 implies that |G| = p3 . The converse is clear by
Fact 3.1.
Theorem 3.3. Let G be a finite nonabelian p-group of maximal class. Then
|Aut(G) : Inn(G)|p = p if and only if either G is of order p3 or one of the groups
of order p4 in Fact 2.7.
Proof. Let G be a p-group of maximal class. Then by Remark 3.2 we have
|G||Aut(G)|. Hence by Lemma 3.2, |Aut(G)|p = |G|. Now the result follows from
Fact 2.7.
Corollary 3.2. Let G be a finite p-group of order p4 . Then |Aut(G) : Inn(G)|p = p
if and only if G is one of the groups (a)–(d) in Fact 2.7.
Proof. It follows from Theorems 3.1 and 3.2 that G is of maximal class. Now
Theorem 3.3 yields the result.
Theorem 3.4. Let G be a finite nonabelian p-group with cyclic Frattini subgroup.
Then |Aut(G) : Inn(G)|p = p if and only if either G is the semidihedral group of
order 16, or G is of order p3 .
Proof. Let G be a group with cyclic Frattini subgroup. By Remark 3.2 we know
that |G| | |Aut(G)|. By Fact 2.9, |Aut(G)|p = |G| if and only if either G is isomorphic
to the semidihedral group of order 16, or Z(G) is cyclic and |G/Z(G)| = p2 . Now
the result follows from Lemma 3.2, Fact 3.1 and Theorem 3.2.
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Corollary 3.3. Let G be a finite nonabelian p-group with noncyclic Frattini subgroup. If |Aut(G) : Inn(G)|p = p then
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AutΦ (G) = Inn(G)CAutΦ (G) (Z(Φ(G))).
Proof. Let Φ(G) be noncyclic. Then by Fact 2.10, we have
CAutΦ (G) (Z(Φ(G))) Inn(G).
Now since Inn(G)CAutΦ (G) (Z(Φ(G))) ≤ AutΦ (G), the result follows.
We turn now to the case of groups of order p5 . Consider the following code in
GAP [21].
f:=function(p,n)
local a,L;
a:=AllSmallGroups(p^n);
L:=Filtered(a,i->Gcd(Size(AutomorphismGroup(i))*Size(Center(i)),
p^(n+2))=p^(n+1));
return(Size(L));
end;
This code accepts a prime number p and a positive integer n. Then it returns the
number of groups G of order pn , in the GAP Small Groups Library, for which
|Aut(G) : Inn(G)|p = p. We have f(2,5)=0, which means there is no such a group
of order 25 . Moreover, one can see that for each prime 3 ≤ p ≤ 47 there is just one
group G of order p5 such that |Aut(G) : Inn(G)|p = p.
Theorem 3.5. Let G be a group of order p5 . Then |Aut(G) : Inn(G)|p = p if and
only if p > 2 and
2
3
G = a, b | ap = 1, bp = 1, [b, a] = bp .
For the sake of clarity, we break the proof into four steps.
Step 1. Let p be an odd prime and let G1 be a group with the following power
commutator presentation,
G1 = g1 , g2 , g3 , g4 , g5 | g4 = [g2 , g1 ], g5 = g3p , g1p = g2p = g4p = g5p = 1,
[g3 , g1 ] = [g5 , g1 ] = [g4 , g2 ] = [g5 , g2 ] = [g4 , g3 ] = 1,
[g4 , g1 ] = g5 = [g3 , g2 ].
Then |G1 | = p5 and |Aut(G1 ) : Inn(G1 )|p > p.
Proof. To show the consistency of the above presentation, it suffices to check that
for each of the following pairs of test words the collections of both words coincide
(see [20, p. 424]).
(a) gk (gj gi ) and (gk gj )gi , for 5 ≥ k > j > i ≥ 1,
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(b) (gjp )gi and gjp−1 (gj gi ), for 5 ≥ j > i ≥ 1,
(c) gj (gip ) and (gj gi )gip−1 , for 5 ≥ j > i ≥ 1 and
(d) (gip )gi and gi (gip ), for 5 ≥ i ≥ 1.
This checking is straightforward. For instance, by induction on l we have
g4 g1l = g1l g4 g5l ,
(l)
g2 g1l = g1l g2 g4l g52 .
Hence,
(p−1)
(g2 g1 )g1p−1 = g1 g2 g4 g1p−1 = g1 g2 g1p−1 g4 g5p−1 = g1p g2 g4p−1 g5 2 g4 g5p−1 = g2 = g2 g1p .
Thus (c) holds in G1 for i = 1, j = 2.
Since the presentation is consistent and G1 has the relative orders (p, p, p, p, p),
the order of G1 is p5 and its nilpotency class is 3. By von Dyck’s theorem, the
mapping
α : g1 → g1 g4 , g2 → g2 , g3 → g3 , g4 → g4 , g5 → g5 ,
defines an automorphism α ∈ IA(G1 )\Inn(G1 ) and it has order p. Moreover, the
mapping
when p > 3,
g1 → g1 g2 , g2 → g2 , g3 → g3 g4−1 , g4 → g4 , g5 → g5
β:
−1
−1
−1
−1
g1 → g1 g2 g3 , g2 → g2 g3 , g3 → g3 g4 , g4 → g4 g5 , g5 → g5 when p = 3,
is a p-automorphism in Aut(G1 )\IA(G1 ). Hence
|Aut(G1 ) : Inn(G1 )|p = |Aut(G1 ) : IA(G1 )|p |IA(G1 ) : Inn(G1 )|p ≥ p2 .
This proves the claim.
Step 2. Let G be a group of order p5 , p > 2, such that |Aut(G) : Inn(G)|p = p.
Then G is two-generated of nilpotency class 3, Z(G) is of order p and G is cyclic
of order p2 .
Proof. It follows from Theorems 3.1–3.3 that G is of nilpotency class 3. By Fact
2.5, G is metabelian. If |G | = p3 then G is two-generated and we may choose
a, b ∈ G such that G = a, b and G/G = aG × bG . Now it follows from
Fact 2.4 that |IA(G)| = p6 . This contradicts the assumption. Thus we may suppose
that |G | = p2 . Since |G : Z(G)| ≤ p4 , |G| | |Aut(G)|, by Remark 3.2. Moreover, we
conclude from Lemma 3.2 that |Z(G)| = p.
Then, we show d(G) = 2. Assume for a contradiction that d(G) = 3. Thus
G = Φ(G) and it is easy to see that Autc (G) is elementary abelian of order p3 .
Since Autc (G) ∩ Inn(G) ∼
= Z2 (G)/Z(G), we get Z2 (G)/Z(G) is elementary abelian
of order p2 . By Theorem 3.4, G is not cyclic. Let u be an element in G \Z(G).
Thus Z2 (G)/Z(G) = uZ(G) × vZ(G), for some v ∈ Z2 (G). Let M = CG (u),
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Finite p-groups with the least number of outer p-automorphisms
N = CG (v). Assume that M = N and let g ∈ G\M . Then Z(G) = [g, u] = [g, v].
Thus [g, u] = [g, v]i , for some 1 ≤ i < p. This means that v i u−1 ∈ Z(G), contrary
to our choice of u, v. Therefore M and N are distinct maximal subgroups of G. Let
a ∈ N \M and b ∈ M \N .
Then by Fact 2.1, the mapping a → au can be extended to an automorphism,
α, of G of order p which fixes M elementwise. Let for some g ∈ G, α = θg ,
the inner automorphism induced by g. Since M and N are distinct, it follows
that CG (Z2 (G)) = CG (u) ∩ CG (v) = Z2 (G). Because α|M = idM , we get g ∈
CG (M ) ⊆ Z2 (G). Thus [a, g] = u ∈ Z(G), which is a contradiction. Therefore
α ∈ IA(G)\Inn(G) is of order p.
If v p = 1, then let β be the map defined by b → bv. By Fact 2.1, β extends to
an automorphism of G which fixes N elementwise. In fact β is a p-automorphism
in Aut(G)\IA(G). Thus |Aut(G) : Inn(G)|p > p. Therefore we may assume that
v p = 1. Hence Z(G) = v p . By our choices of u and v, [a, v] = 1 = [b, u]. So
[G , b] = 1 and [b, a, a] = v ps , for some 0 < s < p. Moreover, we may assume that
[v, b] = v p . Since sp−1 = 1 mod p, we have
[b, as
Replacing a and v by as
p−1
2
p−1
2
[v, a] = 1,
, as
p−1
2
p−1
] = [b, a, a]s
= (v s )p .
and v s , respectively, we get
[v, b] = v p = [b, a, a],
[b, a, b] = 1.
(3.1)
Since G = [a, b], v p , we have ap = [a, b]i v pj , for some 0 ≤ i, j < p. If i = 0,
then it follows that [a, b] ∈ ap v −pj . Hence [a, b] ∈ Z(G), a contradiction. Thus
ap = v pi , for some 0 ≤ i < p. In addition, by Fact 2.6, abp = bp a. Thus bp ∈ Z(G)
and hence bp = v pj for some 0 ≤ j < p. Therefore, replacing a and b by av −i and
bv −j , respectively, we may assume that ap = 1 = bp and the relations of (3.1) hold
as well.
Letting g1 = a, g2 = b, g3 = v, we see that G has the power commutator
presentation of G1 in Step 1, which is a contradiction. This shows that d(G) = 2.
Now, assume that G is elementary abelian. Then by Fact 2.6, ap , bp ∈ Z(G).
Since |Z(G)| = p, we have Z(G) ≤ G . Thus G = Φ(G), a contradiction. Therefore
G is cyclic and the proof is complete.
Step 3. Let G be a group of order p5 , p > 2 such that |Aut(G) : Inn(G)|p = p.
Then G is generated by two elements a and b, which satisfy the following relations:
2
ap = 1,
3
bp = 1
and [b, a] = bp .
(∗)
Proof. By the previous step, we have G/G = aG × bG , for some a, b ∈ G
such that |aG | = p2 and |bG | = p. Since G is of nilpotency class 3, by Fact 2.6,
p
we have [bp , a] = [b, a]p [b, a, b](2) = [b, a]p . Thus bp cannot be an element of Z(G).
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A. Abdollahi, M. Ahmadi & S. M. Ghoraishi
Therefore G = bp and
[b, a, b] = 1.
(3.2)
2
Similar argument shows that ap ∈
/ Z(G) and obviously ap < G , therefore
3
3
p2
p2 i
p ≤ |a| ≤ p . If |a| = p then a = b , for some 0 < i < p. Thus by Fact 2.6, one
2
can see that (ab−i )p = 1 and [b, ab−i , b] = 1. Replacing a by ab−i , we can assume
without loss that
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2
2
ap = 1.
(3.3)
We have Z(G) = [b, a]p = [b, a, a]. Thus for some 0 < i < p, [b, a]p = [b, a, a]i .
Therefore by Fact 2.6 we have
2
[b, a]pi = [b, a, a]i ⇒ [b, ai ]p = [b, ai , ai ].
Hence, replacing a by ai we can assume
[b, a]p = [b, a, a].
(3.4)
By the assumption bp = [b, a]j for some j, where gcd(j, p) = 1. Thus
j
−p( )+j
j
j
2
[b, a−p(2)+j ] = [b, a]−p(2)+j+p( 2 ) = [b, a]j = bp
and
j
j
j
2
[b, a−p(2)+j , a−p(2)+j ] = [bp , a−p(2)+j ] = bp .
j
Replacing a by a−p(2)+j , we get
[b, a] = bp ,
(3.5)
and the relations (3.2)–(3.5) still hold. Therefore G = a, b, where a and b
satisfy (∗).
Step 4. Let p be a prime greater than 2 and let G be a group with the following
presentation,
2
a, b | ap = 1,
3
bp = 1,
[b, a] = bp ,
2
[bp , a] = 1.
Then G has order p5 and |Aut(G) : Inn(G)|p = p.
Proof. First we show that a group with the above presentation exists. Let x
and y be cyclic groups of orders p2 and p3 , respectively. Then α : y → y 1+p
is an automorphism of y of order p2 . Define ϕ : x → α by x → α and let
H = x ϕ y, the semidirect product of y by x with respect to ϕ. By von
Dyck’s theorem G is isomorphic to H and hence |G| = p5 .
a → abpj
Now we show that every automorphism of G is of the form b → apk bl , where
0 ≤ j ≤ p2 − 1, 0 ≤ k ≤ p − 1, 1 ≤ l ≤ p3 − 1 and gcd(l, p) = 1.
→ ai bj
Suppose that the mapping α : ab →
is an automorphism of G. Then the first
ak bl
relation of G implies that α is of the form
a → ai bpj
b → ak bl
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. Since G = bp = (ak bl )p 2nd Reading
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Finite p-groups with the least number of outer p-automorphisms
p
p
and (ak bl )p = akp blp [bl , ak ][bl , bk ](2) [bl , ak , ak ](3) , p divides k. Hence α is of the form
a → ai bpj
. Now by the third relation of G, we have
b → apk bl
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[apk bl , ai bpj ] = (apk bl )p
p
⇔ [bl , ai ] = bpl [bl , apk ](2)
i
⇔ bpl(i+p(2)) = bpl .
This implies that i + p 2i = 1 mod p2 . One can deduce that α is of the form
a → abpj
,
b → apk bl
where 0 ≤ j ≤ p2 − 1, 0 ≤ k ≤ p − 1, 1 ≤ l ≤ p3 − 1 and gcd(l, p) = 1.
On the other hand, by von Dyck’s theorem every such a map determines an
automorphism of G. Therefore the total number of automorphisms of G is p5 (p−1).
Since G is of order p5 and Z(G) is of order p, the result follows.
Corollary 3.4. Let G be a finite nonabelian p-group such that |G : Z(G)| ≤ p4 .
Then |Aut(G) : Inn(G)|p = p if and only if G is of order at most p5 and one of the
groups of Theorems 3.2, 3.3 or 3.5.
Proof. Let G be a finite p-group such that |G : Z(G)| ≤ p4 . Then it follows from
Remark 3.2 that |G| | |Aut(G)|. By Lemma 3.2, |Z(G)| = p. Thus |G| ≤ p5 and the
result follows.
Acknowledgments
The authors thanks the editor and referee for their valuable comments which
improved the presentation of the paper. The first author was supported in part
by grant No. 94050219 from School of Mathematics, Institute for Research in Fundamental Sciences (IPM). The first author was additionally financially supported
by the Center of Excellence for Mathematics at the University of Isfahan.
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