Order problem in nilpotent groups

Let $G$ be a f.g. nilpotent group. I wanted to know if the order problem (given $g \in G$, deciding if there exists $n$ s.t. $g^n=e$) is decidable in $G$? In such a group, the word problem is decidable and we know from Blackburn that so is the conjugacy problem (deciding if $g$ and $h$ … Read more

Uniform word problem in finitely presented simple groups

The following question arose in the comments on this question, and it seems like a reasonable question to ask in its own right. I’ve added some additional details. The word problem in any fixed finitely presented simple group G is decidable; this is Kuznetsov’s theorem. However, the procedure by which one solves it is not … Read more

Generating a group by randomly sampling generators

Let G be a finite abelian group, n a positive integer and let Gn denote the direct product of n copies of G. We say an element of Gn is full if it acts as a nonidentity element of G in each of the factors of Gn. Now consider the following random process. Sample a … Read more

Conjugacy in right-angled Artin groups

I am looking for a reference containing the following result: Let a and b be two elements of a right-angled Artin group A. Assume that a and b have minimal length (with respect to the canonical generating set of A) in their conjugacy classes. Let a1⋯an and b1⋯bm be words of minimal length representing a … Read more

Can one reduce to ‘reversing’ the right multiplier finite-state automata of an automatic group to obtain a biautomatic structure?

Let (G,A,W,{Ra}a∈A∪{1}) be a group equipped with an automatic structure, where G is the group, A is a finite set of generators of G, W is the word-acceptor finite-state automaton, and Ra is the right multiplier finite-state automaton for a∈A∪{1}. Recall that Ra accepts (up to padding by a symbol which I’ll denote by p) … Read more

Complementing the red and blue boolean cube?

Given a boolean $0/1$ cube in $n$ dimensions with $2^{n-1}$ red and $2^{n-1}$ blue points can we complement the cube (blue becomes red and vice versa) in $\operatorname{poly}(n)$ transformations? Here by transformation I mean the following. Cutting the cube by $h=\operatorname{poly}(n)$ hyperplane inequalities each describable by $poly(n)$ bit coefficients into $r=\operatorname{poly}(n)$ pieces $P_1,\dots,P_r$. Permuting coordinates … Read more

Minimal generation for finite abelian groups

Let G be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups: 1) With orders d1,d2,…,dk in such a way that di|di+1, 2) With orders that are powers of not necessarily distinct primes pα11,…,pαnn. Is it true, and how can one prove that the cardinality … Read more

A question about generating set of groups and epimorphism

Do there exist non-isomorphic finitely generated groups, G and H, along with epimorphisms ϕ:G→H and ψ:H→G, such that every generating set of these groups is an image of a generating set, that’s mean, for every generating sets {g1,…,gm} and {h1,…,hn} of G and H respectively, there are generating sets {y1,…,ym} and {x1,…,xn} such that gi=ψ(yi),andhj=ϕ(xj) … Read more

Cyclic subgroups of finite abelian groups

I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one. Are there simple formulas if one restricts to low rank for the subgroups? For example, are there formulas for enumerating cyclic subgroups, or subgroups whose minimal number … Read more

Computations with conetypes of hyperbolic groups

I’d like to know if there exists (and, in this case, where I can find it) some computer program/programming language/any kind of software that can find explicitly the conetypes of a hyperbolic group on which I am working (a presentation of which is known). Answer My KBMAG package can compute a finite state automaton that … Read more