Volume 22, Issue 3

Each GGS-group on the p -adic tree is defined by a (p-1){{(p-1)}}-tuple of integers modulo p , called the defining tuple. It is proved that two GGS-groups are conjugate (equivalently, by a theorem of Grigorchuk and Wilson, isomorphic) in the automorphism group of the tree if and only if their defining tuples can be transformed into each other by multiplication by an integer modulo p and performing certain reorderings of the tuple. Finally, the number of isomorphism classes of GGS-groups is calculated.

Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups, so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then prove that torsion-free pseudo-fibered groups satisfy a Tits alternative. We conclude by proving that a purely hyperbolic pseudo-fibered group acts on the 2-sphere as a convergence group. This leads to an interesting question if there are examples of pseudo-fibered groups other than 3-manifold groups.

For each symmetric, aperiodic probability measure μ on a finitely generated group G , we define a subset Aμ{A_{\mu}} consisting of group elements g for which the limit of the ratio μ∗n⁢(g)/μ∗n⁢(e){{\mu^{\ast n}(g)}/{\mu^{\ast n}(e)}} tends to 1. We prove that Aμ{A_{\mu}} is a subgroup, is amenable, contains every finite normal subgroup, and G=Aμ{G=A_{\mu}} if and only if G is amenable. For non-amenable groups we show that Aμ{A_{\mu}} is not always a normal subgroup and can depend on the measure. We formulate some conjectures relating Aμ{A_{\mu}} to the amenable radical.

The group of formal power series under substitution over 𝔽p{\mathbb{F}_{p}}, the so-called Nottingham group, is a pro- p group so it is a metric space. The intertwining of these two structures is an important object of study. This paper is concerned with how the power map influences the distance of elements. Suppose that d⁢(f,g)=d1{d(f,\,g)=d_{1}} while d⁢(f,1)=d2≥d1{d(f,1)=d_{2}\geq d_{1}}. In this paper we provide a sharp bound for d⁢(fj,gj){d(f^{j},\,g^{j})} in terms of d1,d2{d_{1},\,d_{2}} and the exponent j . This bound confirms a conjecture by K. Keating.

For any compact, connected Lie group G and any finitely generated nilpotent group Γ, we determine the cohomology of the path component of the trivial representation of the group character variety (representation space) Rep⁢(Γ,G)1{{\rm Rep}(\Gamma,G)_{1}}, with coefficients in a field 𝔽{{\mathbb{F}}} with characteristic 0 or relatively prime to the order of the Weyl group W . We give explicit formulas for the Poincaré series. In addition, we study G -equivariant stable decompositions of subspaces X⁢(q,G){{\rm X}(q,G)} of the free monoid J⁢(G){J(G)} generated by the Lie group G , obtained from representations of finitely generated free nilpotent groups.

For the group of endo-permutation modules of a finite p -group, there is a surjective reduction homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic p . We prove that this reduction map always has a section which is a group homomorphism.

A group G is said to be (l,m,n){(l,m,n)}-generated if it can be generated by two suitable elements x and y such that o⁢(x)=l{o(x)=l}, o⁢(y)=m{o(y)=m} and o⁢(x⁢y)=n{o(xy)=n}. In [J. Moori, (p,q,r){(p,q,r)}-generations for the Janko groups J1{J_{1}} and J2{J_{2}}, Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes (p,q,r){(p,q,r)} for which a finite non-abelian simple group is (p,q,r){(p,q,r)}-generated. In the present article, we partially answer this question for Fischer’s largest sporadic simple group Fi24′{\mathrm{Fi}_{24}^{\prime}} by determining all (3,q,r){(3,q,r)}-generations, where q and r are prime divisors of |Fi24′|{\lvert\mathrm{Fi}_{24}^{\prime}\rvert} with 3<q<r{3<q<r}.

We prove that if G is a finite group, then the exponent of its Bogomolov multiplier divides the exponent of G in the following four cases: (i) G is metabelian, (ii) exp⁡G=4{\exp G=4}, (iii) G is nilpotent of class ≤5{\leq 5}, or (iv) G is a 4-Engel group.

In this note, we prove that D8×C2n-3{D_{8}\times C_{2}^{n-3}} is the non-elementary abelian 2-group of order 2n{2^{n}}, n≥3{n\geq 3}, whose number of subgroups of possible orders is maximal. This solves a conjecture by Haipeng Qu. A formula for counting the subgroups of an (almost) extraspecial 2-group is also presented.

We study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form T×H{T\times H}, where T is a 2-group and H is a group of odd order. This includes all nilpotent and hence abelian groups.

A quasi-antichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasi-antichain is the number of atoms. For a positive integer w≥3{w\geq 3}, a quasi-antichain of width w is denoted by ℳw{\mathcal{M}_{w}}. In [B. Brewster, P. Hauck and E. Wilcox, Quasi-antichain Chermak–Delgado lattice of finite groups, Arch. Math. 103 2014, 4, 301–311], it is proved that ℳw{\mathcal{M}_{w}} can be the Chermak–Delgado lattice of a finite group if and only if w=1+pa{w=1+p^{a}} for some positive integer a and some prime p . Let t be the number of abelian atoms in 𝒞⁢𝒟⁢(G){\mathcal{CD}(G)}. In this paper, we completely answer the following question: which values of t are possible in quasi-antichain Chermak–Delgado lattices?