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No, it's not possible. For notational convenience assume $S$ is symmetric and $1\in S$, so that $B_{G,S}(n) = S^n$. Suppose $$ |H\cap S^{n_i}|/|S^{n_i}|\to 1 $$ for some subsequence $(n_i)$. Let $x$ …

No, one cannot. For instance the dihedral group $D_8$ has centre $Z(D_8)\cong C_2$ and $D_8/Z(D_8)\cong C_2\times C_2$, but $D_8\ncong C_2\times C_2\times C_2$. I might paraphrase your argument as fo …

No. Let $G = \langle x, y \rangle$ be a $2$-generated non-free group with $x$ of infinite order such that $F = \langle x, g\rangle \cong F_2$ for some $g \in G$. Then we get a counterexample by taking …

Try not to be seduced by balls. When it comes to infinite groups, you get much better theory if you focus on random walks. For example, the most basic thing you might ask of your density function is t …

Here I will follow the notation from Dixon--Mortimer, rather than OP's, because it seems to be more standard, and clearly distinguishes the full and finitary symmetric groups. Let $\Omega$ be an infin …

Q0: For references try starting with Robinson's A course in the theory of groups, starting around 14.5.5. There the FC center is defined and some characterizations are given for FC groups (groups with …

Not a complete answer: Let $G = \prod_F F$ be the direct product of all finite groups and let $F_\omega$ be the free group on a countably infinite collection of generators. Obviously $G$ and $F_\omega …

$\newcommand{\op}[1]{\operatorname{#1}}\def\GL{\op{GL}}$I am interested in the subgroups of $\GL(n,K)$ of the form $\GL(n/s, E)$ for $E$ some extension field of $K$ of degree $s$. These arise in the f …

This question is inspired by Probability of commutation in a compact group, which asked whether $P(xyx^{-1}y^{-1} = 1)$ could take values strictly between $0$ and $1$ on a compact connected group. Tha …

Given a finitely generated group $G$ and a finite subset $S$, let $\omega_G(S) = \inf_{n\geq1} |S^n|^{1/n}$ (this is the exponential growth rate of $S$). Is there a finitely generated group $G$ w …

For which $n$ is there a unique perfect group of order $n$? Are there infinitely many such $n$? Some guesses for infinite sequences of such $n$: $|\mathrm{PSL}(2,p)|$, $|\mathrm{SL}(2,p)|$, $|A_m|$, …

There must be no other subgroup of $H$ isomorphic to $K$. Certainly this property implies your property, as $g^{-1}Kg$ is always isomorphic to $K$. Conversely if $K'\leq H$ is isomorphic to $K$ let $ …

Your argument for three exponentials can be simplified a bit by using the multiplicative version of van der Corput instead of the additive version. Specifically, if your equation $$2^{y^{2^k}} = 2^{y} …

I have no reference for this problem, but let's at least write down the trivial bounds. Let $s_1,\dots s_r\in S^n$ and suppose $n>|S|^r$. Associate to each index $i$ the element $$(\pi_i(s_1),\dots,\ …

Let $R$ be a ring. An elementary matrix over $R$ is a matrix with $1$s along the diagonal and at most one other nonzero entry. Let $\text{EL}_n(R)$ denote the subgroup of $\text{GL}_n(R)$ generated by …