Volume 8, Issue 4

We classify the groups which are certain split extensions of special 2-groups of the form 2 3+3 m , m  ≥ 1, by the group L 3 (2). These groups behave very much like extraspecial 2-groups and we call them tri-extraspecial groups. A tri-extraspecial group of this form exists if and only if m is a positive even integer, and for every n  ≥ 1 there are exactly two tri-extraspecial groups of the form 2 3+6 n :  L 3 (2). We denote these groups by T + (2 n ) and T − (2 n ). Let ε be + or −. Then the isomorphism type of Q  (2 n ) ≔  O 2 ( T  ε (2 n )) is independent of ε . The automorphism group A ε (2 n ) of T  ε (2 n ) is a non-split extension of Q  (2 n ) by the direct product L 3  (2) ×  S 2 n  (2) × 2. The group A ε (2 n ) permutes transitively the conjugacy classes of L 3 (2)-complements to Q  (2 n ) in T  ε (2 n ). If  S  ε (2 n ) is the stabilizer in A ε (2 n ) of one of these classes of complements, then S  ε (2 n ) is a split extension of Q  (2 n ) by L 3 (2) × (2). The group S  ε (2 n ) is isomorphic to the stabilizer in the orthogonal group (2) of a 3-dimensional totally singular subspace in the natural module. Even more remarkably, a subgroup of A + (4) which is a non-split extension of Q  (4) by L 3  (2) ×  S 5 is the so-called pentad subgroup in the fourth sporadic simple group of Janko J 4 , while a subgroup of index 2 in A − (4) which is a non-split extension of Q  (4) by L 3  (2) ×  S 6  ≅ L 3  (2) ×  S 4 (2) is a maximal 2-local subgroup in the largest Fischer 3-transposition group Fi 24 . The two sporadic examples were the primary motivation for our interest in tri-extraspecial groups.

For each prime p , we determine the smallest integer n such that there exists a group of order p n in which the set of commutators does not form a subgroup. We show that n  = 6 for any odd prime and n  = 7 for p  = 2.

A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the small class case. The approach is also used to obtain some recent results on the capability of certain nilpo tent groups of class 2. We also establish a necessary condition for the capability of an arbitrary p  -group of class k , and some further results.

Let G be a finite solvable group and χ , ψ ∈ Irr( G ) be complex characters of G . Let α   be an irreducible constituent of the product χ ψ . We show that the derived length of Ker( α  )  |  Ker(  χψ  ) is bounded by a linear function on the number of distinct irreducible constituents of χψ .

A Coxeter system ( W ,  S  ) is called affine-free if its Coxeter diagram contains no affine subdiagram of rank ≥ 3. Let ( W ,  S  ) be a Coxeter system of finite rank (i.e. | S  | is finite). The main result is that W  is affine-free if and only if W   has finitely many conjugacy classes of reflection triangles. This implies that the action of W   on its Coxeter cubing (defined by Niblo and Reeves [G. Niblo and L. Reeves. Coxeter groups act on CAT(0) cube complexes. J. Group Theory  6 (2003), 399–413]) is cocompact if and only if ( W ,  S  ) is affine-free. This result was conjectured in loc. cit. As a corollary, we obtain that affine-free Coxeter groups are biautomatic.

We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ  -groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ  -groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n  -dimensional affine space, and show that its relative characteristic is ( n ,  n  ). Then we characterize the irreducible algebraic sets of relative characteristic ( n ,  n  ) and (0,  k  ) where 0 ≤  k  ≤  n . We also obtain some examples of somewhat unusual algebraic sets over ρ  -groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.

Let Γ be a finitely generated infinite group. Denote by K  (Γ ) the FC-centre of Γ, i.e. the subgroup of all elements of Γ having only finitely many conjugates in Γ. Let QI(Γ ) denote the group of quasi-isometries of Γ with respect to a word metric. We prove that the natural homomorphism θ Γ  : Aut(Γ ) → QI(Γ ) is a monomorphism only if K  (Γ ) equals the centre  Z  (Γ ) of Γ. The converse holds if K  (Γ ) =  Z  (Γ ) is torsion-free. When K  (Γ ) is finite we show that  is a monomorphism where  = Γ | K  (Γ ). We apply this criterion to a number of classes of groups arising in combinatorial and geometric group theory.

We describe a linear time probabilistic algorithm to recognize Whitehead minimal elements (elements of minimal length in their automorphic orbits) in free groups of rank 2. For a non-minimal element the algorithm gives an automorphism that is most likely to reduce the length of the element. This method is based on linear regression and pattern recognition techniques.