Volume 10, Issue 3

Let G be a finite non-abelian group satisfying Φ( G ) = 1 and denote by U the nilpotent residual of G . In this paper, we prove that if G is of odd order then , and if G is of even order not divisible by a Mersenne or a Fermat prime then . These results are best possible and the assumption Φ( G ) = 1 cannot be omitted.

An infinite f.g. group G is quasi-finitely axiomatizable (QFA) if there is a first-order sentence ϕ such that G ⊨ ϕ, and if H is a f.g. group such that H ⊨ ϕ, then G ≅ H . The first result is that all Baumslag–Solitar groups of the form 〈 a, d | d -1 ad = a m 〉 are QFA. A f.g. group G is a prime model if and only if there is a tuple g 1 , … , g n generating G whose orbit (under the automorphisms of G ) is definable by a first-order formula. The second result is that there are continuum many non-isomorphic f.g. groups that are prime models. In particular, not all are QFA.

We construct a finitely presented torsion-free simple group Σ 0 , acting cocompactly on a product of two regular trees. An infinite family of such groups was introduced by Burger and Mozes [M. Burger and S. Mozes. Finitely presented simple groups and products of trees. C. R. Acad. Sci. Paris Sér. I Math . 324 (1997), 747–752.], [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math . 92 (2001), 151–194.]. We refine their methods and construct Σ 0 as an index 4 subgroup of a group presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math . 92 (2001), 151–194., Theorem 6.4] needs more than 18000 relations, and the smallest simple group constructed in [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math . 92 (2001), 151–194., §6.5] needs even more than 360000 relations in any finite presentation.

A generalized Hantzsche–Wendt (GHW) group is by definition the fundamental group of a flat n -manifold with holonomy group ℤ 2 n -1 , and a Hantzsche–Wendt (HW) group is a GHW group corresponding to an orientable manifold (with n odd). We prove that for any n -dimensional HW group with n > 3, the commutator subgroup and translation subgroup are equal, and hence the abelianization of the group is ℤ 2 n -1 . We also give examples of GHW groups with the same property for all n > 4. These groups are all examples of torsion-free metabelian groups with abelianizations ℤ 2 k for some k > 3.