On finite Alperin groups with abelian second commutator subgroups

B. Veretennikov

J. L. Alperin studied in [1] the groups in which all 2-generated subgroups have a cyclic commutator subgroup. Such groups are called Alperin groups. It was proved in [1] that, for an odd prime \(p\), finite Alperin \(p\)-groups are metabelian; i. e., they have an abelian commutator subgroup. However, finite Alperin 2-groups may be nonmetabelian. For example, a nonmetabelian finite Alperin 2-group with second commutator subgroup of order 2 was constructed in [2], and infinite series of finite Alperin 2-groups with second commutator subgroups of orders 2 and 4 were constructed in [3].

In [4,5], the author proved the existence of finite Alperin 2-groups with cyclic second commutator subgroup of arbitrarily large order and with elementary abelian second commutator subgroup of arbitrary rank.

In this communication, we announce for every finite abelian group \(H\) the existence of a finite Alperin group \(G\) whose second commutator subgroup \(G''\) is isomorphic to \(H\). This result is an easy consequence of the following theorem.

In what follows, a homocyclic group means a group that is isomorphic to \(Z_m \times \dots \times Z_m\) where \(m\) is a positive integer.

Theorem 1. Let \(m\), \(n\) be positive integers, \(n \geq 3\), and let a group \(G\) be defined by the generators \(a_i, f_{ij}, \tau_{ijk},\) where \(1 \leq i,j,k \leq n\), and the defining relations

1) \(a_i^2 = 1\),
2) \([a_i,a_j]=f_{ij}\),
3) \([f_{ij},a_k]=f_{jk}^2 f_{ki}^2 \tau_{ijk}^{-2}\),
4) \([\tau_{ijk},a_s]=1\),
5) \(f_{ij}^{4m}=1\),
6) \(\tau_{ijk}^{m}=1\),
7) \([f_{ij},f_{ks}]=\tau_{kjs} \tau_{ksi}\),
8) \((f_{ij} f_{jk} f_{ki})^4=\tau_{ijk}\),
9) \(\tau_{sij} \tau_{sjk} \tau_{ski} = \tau_{ijk}\),

for all positive integers \(i,j,k,s \in [1,n]\). Then the following statements are valid:

I) The group \(G\) has a normal series: \[1 < \langle \tau_{ijk} | 1 \leq i,j,k \leq n \rangle = G'' < G'' \langle f_{12} \rangle < G'' \langle f_{12} \rangle \langle f_{13} \rangle < \dots < \] \[ G'' \langle f_{12} \rangle \langle f_{13} \rangle \dots \langle f_{1n} \rangle = H < H \langle f_{23} \rangle < H \langle f_{23} \rangle \langle f_{24} \rangle < \dots < \] \[ H \langle f_{23} \rangle \langle f_{24} \rangle \dots \langle f_{n-1,n} \rangle = G' < G' \langle a_{1} \rangle < G' \langle a_{1} \rangle \langle a_{2} \rangle < \dots < \] \[ G' \langle a_{1} \rangle \langle a_{2} \rangle \dots \langle a_{n} \rangle = G,\] in which the first factor-group has order \(m^{\binom{n-1}{2}}\), the following \(n-1\) factor-groups are cyclic of order \(4m\), then the following \(\binom{n-1}{2}\) factor-groups are cyclic of order \(4\), and the last \(n\) factor-groups are cyclic of order \(2\);

II) \(d(G)=n\); \(G=\langle a_1, \dots, a_n \rangle\); \(|a_i|=2\) for any \(i \in [1,n]\); \(|G|=m^{\binom{n}{2}} 2^{n^2}\); \(d(G')=\binom{n}{2}\); \(G'=\langle f_{ij} | 1 \leq i < j \leq n \rangle\); \(G''=\prod\limits_{2 \leq i < j \leq n} \langle \tau_{1ij} \rangle \), a homocyclic group with period \(m\) and rank \(\binom{n-1}{2}\); \(G'' \leq Z(G)\);

III) G is an Alperin group; moreover, for any \(x,y \in G\), at least one of the following equalities is true: \([x,y,y]=1\) or \([x,y,y]=[x,y]^{-2}\).

The following theorem is easily deduced from Theorem 1.

Theorem 2. For any finite abelian group \(H\), there exists a finite Alperin group \(G\) generated by involutions, whose second commutator subgroup \(G''\) is isomorphic to \(H\).

References

1. J. L. Alperin, On a special class of regular \(p\)-groups, Trans. Am. Mat. Soc., 106 N1 (1963), 77–99.
2. B. M. Veretennikov, A conjecture of Alperin, Sib. Mat. Zh., 21 (1980), 200–202.
3. B. M. Veretennikov, On finite 3-generated 2-groups of Alperin, Sib. Elektron. Mat. Izv., 4 (2007), 155–168.
4. B. M. Veretennikov, Finite Alperin 2-groups with cyclic second commutants, Algebra and Logic, 50, N3 (2011), 226–244.
5. B. M. Veretennikov, On finite Alperin 2-groups with elementary abelian second commutants, Sib. Mat. Zh., 53, N3 (2012), 431–443.

See also the author's pdf version: