POWER-COMMUTATOR PRESENTATIONS FOR INFINITE SEQUENCES OF
3-GROUPS
DANIEL C. MAYER
Abstract. For certain infinite sequences of 3-groups G with derived length 2 ≤ dl(G) ≤ 3,
and either with G/G0 ' (3, 3), coclass cc(G) ≥ 1, or with G/G0 ' (9, 3), coclass cc(G) ≥ 3,
parametrized pc-presentations are given in dependence on the class c = cl(G).
1. Preliminaries
For the detailed description of p-groups G with abelianization G/G0 of type (p, p) or (p2 , p),
we need two advanced invariants which coincide for any p-group G and for its metabelianization
G/G00 [18, Thm. 1.1, p. 402].
Definition 1.1. Let G be a p-group of generator rank d(G) = d, put m =
be the maximal (normal) subgroups (of index p) in G, and denote by
pd −1
p−1 ,
let H1 , . . . , Hm
Ti : G/G0 → Hi /Hi0 , gG0 7→ Ti (gG0 )
the Artin transfer from G to Hi , for 1 ≤ i ≤ m, [17, (4), p. 470].
The family κ(G) = (ker(Ti ))1≤i≤m is called the transfer kernel type, TKT, of G, and the family
τ (G) = (Hi /Hi0 )1≤i≤m is called the transfer target type, TTT, of G.
In this article, we shall be concerned with 3-groups G of particular TKTs, resp. pTKTs
(punctured), with d = 2, m = 4, specified at the beginning of each section. By the symbol
∼ we indicate that some of the TKTs are equivalent, since they generate the same orbit under
action of the symmetric group S4 [17].
Definition 1.2. An infinite sequence of p-groups (Gj )j≥0 is called a periodic sequence or coclass
family if its members
(1) share a common coclass cc(Gj ) = r ≥ 1,
(2) are descendants of a common ancestor M0 , which is the root of a coclass tree Tr (M0 ),
(3) share a common TKT κ(Gj ),
(4) share m − 1 common components of the TTT τ (Gj ), whereas the single remaining component depends on the nilpotency class,
(5) share a common parametrized pc-presentation Gj = hx1 , . . . , xd | Rcj (x1 , . . . , xd )i having
the nilpotency class cj = cl(Gj ) as the only parameter.
Remark 1.1. The mainline of the coclass tree Tr (M0 ) consists of one or more periodic sequences
(Mj )j≥0 , whose parametrized pc-presentations approach the pro-p presentation of their projective
limit L = lim j≥0 Mj when the class cj = cl(Mj ) tends to infinity.
←−
In the sequel we characterize 3-groups by their identifer in the SmallGroups library [4] and their
descendants of order bigger than 37 by the notation used in the ANUPQ package [14] of GAP [13]
and MAGMA [15].
Date: March 19, 2014.
2000 Mathematics Subject Classification. Primary 20D15, 20F12, 20F14.
Key words and phrases. power-commutator presentations, 3-groups of derived lengths 2 and 3, central series,
lattice of normal subgroups, coclass trees.
Research supported by the Austrian Science Fund (FWF): P 26008-N25.
1
2
DANIEL C. MAYER
2. 3-groups G with G/G0 ' (3, 3)
For coclass cc(G) = 1 and abelianization G/G0 ' (3, 3) we need
• all three cases of TKTs in section a, that is,
a.1, κ = (0, 0, 0, 0),
a.2, κ = (1, 0, 0, 0), and
a.3, κ = (2, 0, 0, 0) ∼ (3, 0, 0, 0) ∼ (4, 0, 0, 0).
2.1. Periodic 3-groups on coclass tree T1 (h9, 2i). As a familiar entrance, we begin by showing
that certain 3-groups of class at least 5 on the coclass tree T1 (h9, 2i) belong to 4 + 3 = 7 periodic
coclass sequences with period length 2.
Theorem 2.1. For each integer c ≥ 5, there are 4 metabelian descendants G of h9, 2i, having
nilpotency class cl(G) = c, coclass cc(G) = 1, and order |G| = 3c+1 , with two generators x, y and
parametrized pc-presentation
G = h x, y, s2 , s3 , s4 , . . . , sc |
s2 = [y, x], sj = [sj−1 , x] for 3 ≤ j ≤ c,
s3j = s2j+2 sj+3 for 2 ≤ j ≤ c − 3, s3c−2 = s2c ,
R(x) = 1, R(y) = 1 i,
where the relators R(x) and R(y) are given by
(
(1)
R(x)
(2)
R(y)
=
=
x3
for G
x3 s−1
for G
c

3 −2 −1

y s3 s4
−1 −1
y 3 s−2
3 s4 sc

 3 −2 −1 −2
y s3 s4 sc
of TKT a.1 or a.3,
of TKT a.2,
for G of TKT a.1 or a.2,
or
for G of TKT a.3.
For odd class c ≥ 5 the 4 groups are pairwise non-isomorphic σ-groups.
For even class c ≥ 6, the pair of groups sharing the same TKT (a.3) is isomorphic, and thus only
3 groups are pairwise non-isomorphic, and only the mainline group with TKT a.1 is a σ-group.
Remark 2.1. The presentations in Theorem 2.1 are similar to but not identical with Blackburn’s
well-known presentations [5]. See also [17, § 2, pp. 469–470].
Remark 2.2. Pro-3 presentations for the projective limit L of the mainline of coclass tree
T1 (h9, 2i) are given in two different forms by Eick and Feichtenschlager [9, § 9.2, p. 11] resp.
[12, App. A, pp. 90–91]:
(1) either for L as an extension of γ3 (L) ' Z23 by L/γ3 (L),
−2
L = h x, y, s2 , t1 , t2 | [y, x] = s2 , [s2 , x] = t1 , [t1 , x] = t2 , [t2 , x] = t−4
1 t2 ,
−1
x3 = 1, y 3 = t21 t2 , s32 = t−3
1 t2 i,
(2) or for L as an extension of γ4 (L) ' Z23 by L/γ4 (L),
L =
−2
h x, y, s2 , s3 , t1 , t2 | [y, x] = s2 , [s2 , x] = s3 , [s3 , x] = t2 , [t1 , x] = t32 , [t2 , x] = t−2
1 t2 ,
−1
3
x3 = 1, y 3 = s23 t2 , s32 = t−1
1 t2 , s3 = t1 i.
POWER-COMMUTATOR PRESENTATIONS FOR INFINITE SEQUENCES OF 3-GROUPS
3
For coclass cc(G) ≥ 2 and abelianization G/G0 ' (3, 3) we need
• all four cases of TKTs in section E, that is,
E.6, κ = (1, 1, 2, 2),
E.14, κ = (3, 1, 2, 2) ∼ (4, 1, 2, 2),
E.8, κ = (1, 1, 3, 4), and
E.9, κ = (3, 1, 3, 4) ∼ (4, 1, 3, 4),
• both TKTs in section c, that is,
c.18, κ = (0, 1, 2, 2), and
c.21, κ = (0, 1, 3, 4),
• TKT H.4, κ = (2, 1, 2, 2),
• TKT G.16, κ = (2, 1, 3, 4).
2.2. Periodic 3-groups on coclass tree T2 (h243, 6i). Now we show that certain 3-groups of
class at least 5 on the coclass tree T2 (h243, 6i) belong to 6 + 4 = 10 periodic coclass sequences
with period length 2.
Theorem 2.2. For each integer c ≥ 5, there are 6 metabelian descendants G of h243, 6i, having
nilpotency class cl(G) = c, coclass cc(G) = 2, and order |G| = 3c+2 , with two generators x, y and
parametrized pc-presentation
G = h x, y, s2 , t3 , s3 , s4 , . . . , sc |
s2 = [y, x], t3 = [s2 , y], sj = [sj−1 , x] for 3 ≤ j ≤ c,
s3j = s2j+2 sj+3 for 2 ≤ j ≤ c − 3, s3c−2 = s2c , t33 = 1,
R(x) = 1, R(y) = 1 i,
where the relators R(x) and R(y) are given by
(3)
R(x)
=
(4)
R(y)
=
(
x3
for G
3 −1
for G
x sc

−2
−1
3

y s3 s4
−1
y 3 s3−2 s−1
4 sc

 3 −2 −1 −2
y s3 s4 sc
of TKT c.18 or H.4,
of TKT E.6 or E.14,
for G of TKT c.18 or E.6,
or
for G of TKT H.4 or E.14.
For odd class c ≥ 5 the 6 groups are pairwise non-isomorphic σ-groups.
For even class c ≥ 6, the two pairs of groups sharing the same TKT (H.4 and E.14) are isomorphic,
and thus only 4 groups are pairwise non-isomorphic, and only the mainline group with TKT c.18
is a σ-group.
4
DANIEL C. MAYER
Proof. G is a metabelian 3-group with abelian commutator subgroup G0 = hs2 , t3 , s3 , s4 , . . . , sc i.
Due to the nilpotency relation [sc , x] = 1, G is of class cl(G) = c. Since t3 is not contained in the
subgroup hs3 , s4 , . . . , sc i and the centre of G is given by ζ1 (G) = ht3 , sc i, the lower central series
of G has a single bicyclic factor γ3 = ht3 , s3 , γ4 (G)i and G is of coclass cc(G) = 2, and thus of
order |G| = 3c+2 .
The four maximal subgroups of G are given by H1 = hy, G0 i, H2 = hx, G0 i, H3 = hxy, G0 i,
H4 = hxy 2 , G0 i.
For the investigation of the transfers
(
g 3 Hi0
if g ∈ G \ Hi ,
Ti : G/G0 → Hi /Hi0 , gG0 7→
g S3 (h) Hi0 if g ∈ Hi ,
where S3 (h) = 1 + h + h2 ∈ Z[G] for some h ∈ G \ Hi , generators of the derived subgroups
Hi0 , 1 ≤ i ≤ 4, must be determined. Since G0 is a normal subgroup of index 3 in each maximal
subgroup, we obtain Hi0 = [G0 , Hi ] = (G0 )gi −1 when Hi = hgi , G0 i, and some commutator calculus
yields
H10
=
ht3 i,
H20
H30
H40
=
hs3 , s4 , . . . , sc i,
=
hs3 t3 , s4 , . . . , sc i,
=
hs3 t23 , s4 , . . . , sc i.
Now we can calculate the transfer kernels. For this purpose we represent the elements g ∈ G
in the form g ≡ xj y ` mod G0 and solve the congruence Ti (gG0 ) ≡ 1 mod Hi0 .
First we derive expressions for the transfer images of the generators x, y. Since x, y ∈
/ Hi and
s4 , sc ∈ Hi0 for i = 3, 4, we have Ti (xG0 ) ≡ x3 ≡ 1 mod Hi0 and Ti (yG0 ) ≡ y 3 ≡ s23 mod Hi0 for
i = 3, 4.
Further, since y ∈
/ H2 and s3 , s4 , sc ∈ H20 , we have T2 (yG0 ) ≡ y 3 ≡ 1 mod H20 , and since
x∈
/ H1 , we have T1 (xG0 ) ≡ x3 mod H10 .
However, for the action of trace elements as symbolic exponents we need [17, eqn. (6), p. 486].
−1
Since x ∈ H2 and sc ∈ H20 , we have T2 (x) ≡ xS3 (y) ≡ x3 [x, y]3 [[x, y], y] ≡ 1 · s−3
2 [s2 , y] ≡
−1
−1
−1 −1
mod H20 .
s−2
≡ [y, s2 ]s2 ≡ t−1
3
4 s5 [y, s2 ]
Since y ∈ H1 and since we can prove by induction that s33 s34 s5 = 1, we have T1 (y) ≡ y S3 (x) ≡
3
y [y, x]3 [[y, x], x] ≡ s23 s4 sec s32 s3 ≡ s23 s4 sec s24 s5 s3 ≡ s33 s34 s5 sec ≡ sec mod H10 for some exponent 0 ≤
e ≤ 2.
Consequently, we obtain the following expressions for the transfer images:
T1 (x y G ) ≡
(
mod H10
se`
c
j+e`
sc
mod H10
T2 (xj y ` G0 ) ≡
t−j
3
mod H20 ,
Ti (xj y ` G0 ) ≡
s2`
3
mod Hi0 for i ∈ {3, 4}.
j `
0
if x3 = 1, y 3 = s23 s4 sec ,
if x3 = sc , y 3 = s23 s4 sec ,
POWER-COMMUTATOR PRESENTATIONS FOR INFINITE SEQUENCES OF 3-GROUPS
5
2.3. Periodic 3-groups on coclass tree T2 (h243, 8i). Similarly to the previous section, we now
show that certain 3-groups of class at least 6 on the coclass tree T2 (h243, 8i) belong to 6 + 4 = 10
periodic coclass sequences with period length 2.
Theorem 2.3. For each integer c ≥ 6, there are 6 metabelian descendants G of h243, 8i, having
nilpotency class cl(G) = c, coclass cc(G) = 2, and order |G| = 3c+2 , with two generators x, y and
parametrized pc-presentation
G = h x, y, t2 , s3 , t3 , t4 , . . . , tc |
t2 = [y, x], s3 = [t2 , x], tj = [tj−1 , y] for 3 ≤ j ≤ c,
t3j = t2j+2 tj+3 for 2 ≤ j ≤ c − 3, t3c−2 = t2c , s33 = 1,
R(y) = 1, R(x) = 1 i,
where the relators R(y) and R(x) are given by
(
y 3 s−1
for G
3
R(y) =
3 −1 −1
y s3 tc
for G

3 −1 −2 −1

x t3 t4 t5
−2 −1 −1
3
R(x) =
x t−1
3 t4 t5 tc

 3 −1 −2 −1 −2
x t3 t4 t5 tc
(5)
(6)
of TKT c.21 or G.16,
of TKT E.8 or E.9,
for G of TKT c.21 or E.8,
or
for G of TKT G.16 or E.9.
For odd class c ≥ 7 the 6 groups are pairwise non-isomorphic σ-groups.
For even class c ≥ 6, the two pairs of groups sharing the same TKT (G.16 and E.9) are isomorphic,
and thus only 4 groups are pairwise non-isomorphic, and only the mainline group with TKT c.21
is a σ-group.
Remark 2.3. Eick, Leedham-Green, Newman, and O’Brien [11] have determined the projective
limit L = lim j≥0 Mj of the metabelian mainline (Mj )j≥0 of the coclass tree T2 (M0 ) with root
←−
M0 = h243, 6i, resp. h243, 8i. It is given by the pro-3 presentation
L = h t, a, z
|
2
a3 = z f , [t, ta ] = z, tta ta = z 2 ,
z 3 = 1, [z, a] = 1, [z, t] = 1, i,
where f = 0, resp. f = 1. The centre of L is the cyclic group ζ1 (L) = hzi of order 3.
The mainline vertices of T2 (M0 ) are the σ-groups
M2`
`+2
' L/ht3
i
of order 32`+5 and odd class 2` + 3,
M2`+1
`+2
' L/ht3
`+1
, t3
2`+6
of order 3
for ` ≥ 0.
`+1
(ta )−3
i
and even class 2` + 4,
6
DANIEL C. MAYER
Proof. G is a metabelian 3-group with abelian commutator subgroup G0 = ht2 , s3 , t3 , t4 , . . . , tc i.
Due to the nilpotency relation [tc , x] = 1, G is of class cl(G) = c. Since s3 is not contained in the
subgroup ht3 , t4 , . . . , tc i and the centre of G is given by ζ1 (G) = hs3 , tc i, the lower central series of
G has a single bicyclic factor γ3 = hs3 , t3 , γ4 (G)i and G is of coclass cc(G) = 2, and thus of order
|G| = 3c+2 .
The four maximal subgroups of G are given by H1 = hx, G0 i, H2 = hy, G0 i, H3 = hyx, G0 i,
H4 = hyx2 , G0 i.
For the investigation of the transfers
(
g 3 Hi0
if g ∈ G \ Hi ,
Ti : G/G0 → Hi /Hi0 , gG0 7→
g S3 (h) Hi0 if g ∈ Hi ,
where S3 (h) = 1 + h + h2 ∈ Z[G] for some h ∈ G \ Hi , generators of the derived subgroups
Hi0 , 1 ≤ i ≤ 4, must be determined. Since G0 is a normal subgroup of index 3 in each maximal
subgroup, we obtain Hi0 = [G0 , Hi ] = (G0 )gi −1 when Hi = hgi , G0 i, and some commutator calculus
yields
H10
= hs3 i,
H20
H30
H40
= ht3 , t4 , . . . , tc i,
= ht3 s3 , t4 , . . . , tc i,
= ht3 s23 , t4 , . . . , tc i.
Now we can calculate the transfer kernels. For this purpose we represent the elements g ∈ G
in the form g ≡ y j x` mod G0 and solve the congruence Ti (gG0 ) ≡ 1 mod Hi0 .
First we derive expressions for the transfer images of the generators x, y. Since x, y ∈
/ Hi and
t4 , t5 , tc ∈ Hi0 for i = 3, 4, we have Ti (yG0 ) ≡ y 3 ≡ s3 mod Hi0 and Ti (xG0 ) ≡ x3 ≡ t23 mod Hi0
for i = 3, 4.
Further, since x ∈
/ H2 and t3 , t4 , t5 , tc ∈ H20 , we have T2 (xG0 ) ≡ x3 ≡ 1 mod H20 , and since
0
y∈
/ H1 and s3 ∈ H1 , we have T1 (yG0 ) ≡ y 3 ≡ tεc mod H10 with suitable 0 ≤ ε ≤ 1.
However, for the action of trace elements as symbolic exponents we need [17, eqn. (6), p. 486].
Since y ∈ H2 and t4 , t5 , tc ∈ H20 , we have T2 (y) ≡ y S3 (x) ≡ y 3 [y, x]3 [[y, x], x] ≡ s3 tεc · t32 s3 ≡
2 2
s3 t4 t5 tεc ≡ s23 mod H20 .
Since x ∈ H1 and since we can prove by induction that s33 s34 s5 = 1, we have T1 (x) ≡ xS3 (y) ≡
−1
0
e
2
e −2 −1 −1
3
x [x, y]3 [[x, y], y] ≡ t3 t24 t5 tec t−3
2 t3 ≡ t3 t4 t5 tc t4 t5 t3 ≡ tc mod H1 for some exponent 0 ≤ e ≤ 2.
Consequently, we obtain the following expressions for the transfer images:
(
j `
0
T1 (y x G ) ≡
te`
mod H10
c
mod H10
tj+e`
c
T2 (y j x` G0 ) ≡
s2j
3
Ti (y j x` G0 ) ≡
sj3 t`3
if y 3 = s3 , x3 = t3 t24 t5 tec ,
if y 3 = s3 tc , x3 = t3 t24 t5 tec ,
mod H20 ,
mod Hi0 for i ∈ {3, 4}.
POWER-COMMUTATOR PRESENTATIONS FOR INFINITE SEQUENCES OF 3-GROUPS
7
2.4. Periodic 3-groups on coclass tree T3 (h729, 49i − #2; 1). The following result shows that
certain 3-groups of class at least 6 on the entirely non-metabelian coclass tree T3 (h729, 49i−#2; 1),
belong to 6 + 4 = 10 periodic coclass sequences with period length 2.
Theorem 2.4. For each integer c ≥ 6, there are 6 descendants G of h729, 49i − #2; 1, having
nilpotency class cl(G) = c, coclass cc(G) = 3, order |G| = 3c+3 , and derived length dl(G) = 3,
with two generators x, y and parametrized pc-presentation
G = h x, y, s2 , t3 , s3 , s4 , . . . , sc , u5 |
s2 = [y, x], t3 = [s2 , y], sj = [sj−1 , x] for 3 ≤ j ≤ c,
u5 = [s3 , y] = [s4 , y], [s3 , s2 ] = u25 , t33 = u25 ,
s32 = s24 s5 u5 , s3j = s2j+2 sj+3 for 3 ≤ j ≤ c − 3, s3c−2 = s2c ,
R(x) = 1, R(y) = 1 i,
where the relators R(x) and R(y) are given by equations (3) and (4).
For odd class c ≥ 7 the 6 groups are pairwise non-isomorphic σ-groups.
For even class c ≥ 6, the two pairs of groups sharing the same TKT (H.4 and E.14) are isomorphic,
and thus only 4 groups are pairwise non-isomorphic, and only the mainline group with TKT c.18
is a σ-group.
2.5. Periodic 3-groups on coclass tree T3 (h729, 54i − #2; 3). The following result shows that
certain 3-groups of class at least 6 on the entirely non-metabelian coclass tree T3 (h729, 54i−#2; 3),
belong to 6 + 4 = 10 periodic coclass sequences with period length 2.
Theorem 2.5. For each integer c ≥ 6, there are 6 descendants G of h729, 54i − #2; 3, having
nilpotency class cl(G) = c, coclass cc(G) = 3, order |G| = 3c+3 , and derived length dl(G) = 3,
with two generators x, y and parametrized pc-presentation
G = h x, y, t2 , s3 , t3 , t4 , . . . , tc , u5 |
t2 = [y, x], s3 = [t2 , x], tj = [tj−1 , y] for 3 ≤ j ≤ c,
u5 = [t3 , x] = [t4 , x], [t3 , t2 ] = u5 , s33 = u25 ,
t32 = t24 t5 u5 , t3j = t2j+2 tj+3 for 3 ≤ j ≤ c − 3, t3c−2 = t2c ,
R(y) = 1, R(x) = 1 i,
where the relators R(y) and R(x) are given by equations (5) and (6).
For odd class c ≥ 7 the 6 groups are pairwise non-isomorphic σ-groups.
For even class c ≥ 6, the two pairs of groups sharing the same TKT (G.16 and E.9) are isomorphic,
and thus only 4 groups are pairwise non-isomorphic, and only the mainline group with TKT c.21
is a σ-group.
8
DANIEL C. MAYER
3. 3-groups G with G/G0 ' (9, 3)
For coclass cc(G) ≥ 3 and abelianization G/G0 ' (9, 3) we need
• pTKT C.4, κ = (3, 1, 1; 3),
• two pTKTs in section D,
D.5, κ = (2, 1, 1; 3), and
D.10, κ = (4, 1, 1; 3),
• pTKT d.10, κ = (0, 1, 1; 3),
• pTKT B.2, κ = (1, 1, 1; 3).
3.1. Periodic 3-groups on coclass tree T3 (h729, 13i). Similarly as in the previous sections, we
show that certain 3-groups of class at least 8 on the coclass tree T3 (h729, 13i) belong to 9 + 5 = 14
periodic coclass sequences with period length 2.
Theorem 3.1. For each integer c ≥ 8, there are 9 metabelian descendants G of h729, 13i, having
nilpotency class cl(G) = c, coclass cc(G) = 3, and order |G| = 3c+3 , with two generators x, y and
parametrized pc-presentation
G = h x, y, τ, t2 , s3 , t3 , t4 , . . . , tc |
τ = x3 , t2 = [y, x], s3 = [t2 , x], tj = [tj−1 , y] for 3 ≤ j ≤ c,
t3j = t2j+2 tj+3 for 2 ≤ j ≤ c − 3, t3c−2 = t2c , s33 = 1,
[τ, y] = t4 t25 t6 , R(y) = 1, R(τ ) = 1 i,
where the relators R(y) and R(τ ) are given by
(7)
(8)

3

for G of pTKT d.10 or B.2(1) or B.2(2),
y
3 −1
R(y) =
for G of pTKT C.4(1) or D.5(2) or D.10(1),
y tc

 3 −2
y tc
for G of pTKT C.4(2) or D.5(1) or D.10(2),

3 −1 −2 −2 −1

for G of pTKT d.10 or D.10,
τ s3 t5 t6 t7
−2 −2 −1 −1
R(τ ) =
for G of pTKT B.2(1) or C.4(1) or D.5(1),
τ 3 s−1
t
t
t
t
3 5 6 7 c

 3 −1 −2 −2 −1 −2
for G of pTKT B.2(2) or C.4(2) or D.5(2).
τ s3 t5 t6 t7 tc
For odd class c ≥ 9 the 9 groups are pairwise non-isomorphic σ-groups.
For even class c ≥ 8, the four pairs of groups sharing the same pTKT (B.2, C.4, D.5 and D.10)
are isomorphic, and thus only 5 groups are pairwise non-isomorphic, and only the mainline group
with pTKT d.10 is a σ-group.
POWER-COMMUTATOR PRESENTATIONS FOR INFINITE SEQUENCES OF 3-GROUPS
9
3.2. Periodic 3-groups on coclass tree T4 (h2187, 168i−#2; 7). The following result shows that
certain 3-groups of class at least 8 on the entirely non-metabelian coclass tree T4 (h2187, 168i −
#2; 7), belong to 9 + 5 = 14 periodic coclass sequences with period length 2.
Theorem 3.2. For each integer c ≥ 8, there are 9 descendants G of h2187, 168i − #2; 7, having
nilpotency class cl(G) = c, coclass cc(G) = 4, order |G| = 3c+4 , and derived length dl(G) = 3,
with two generators x, y and parametrized pc-presentation
G = h x, y, τ, t2 , s3 , t3 , t4 , . . . , tc , u5 |
τ = x3 , t2 = [y, x], s3 = [t2 , x], tj = [tj−1 , y] for 3 ≤ j ≤ c,
u5 = [t3 , x] = [t4 , x], [τ, t2 ] = [t3 , t2 ] = u5 , s33 = u25 ,
t32 = t24 t5 u5 , t3j = t2j+2 tj+3 for 3 ≤ j ≤ c − 3, t3c−2 = t2c ,
[τ, y] = t4 t25 t6 , R(y) = 1, R(τ ) = 1 i,
where the relators R(y) and R(τ ) are given by equations (7) and (8).
For odd class c ≥ 9 the 9 groups are pairwise non-isomorphic σ-groups.
For even class c ≥ 8, the four pairs of groups sharing the same pTKT (B.2, C.4, D.5 and D.10)
are isomorphic, and thus only 5 groups are pairwise non-isomorphic, and only the mainline group
with pTKT d.10 is a σ-group.
10
DANIEL C. MAYER
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