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Here is a proof for finite perfect groups. Possibly this is already what Dave Benson had in mind. Let $G$ be a finite perfect group. If $G \cong G_1 \times G_2$ for nontrivial $G_1$ and $G_2$ then by …

Let $G$ be a finitely generated group. Can $G$ be embedded as a finite-index subgroup of a 2-generated group? 100-generated? I strongly doubt it but I don't know a counterexample.

As Nick Gill points out, one can certainly have $|B| > |A|$ if you do not assume coprimality. If you assume coprimality then yes $|B| < |A|$. If $A_p$ is the Sylow $p$-subgroup of $A$ then $\DeclareMa …

Repeating from the comments section: This (natural and beautiful) question was previously asked and answered on this site. See Collapsible group words. It also appeared recently on math.se. The questi …

If $G$ is a group and $g_1, g_2 \in G$ let us write $g_1 \sim g_2$ if there is an automorphism $\alpha \in \operatorname{Aut}(G)$ such that $g_1^\alpha = g_2$. Let $F = F_2$ be the free group on two g …

Yes. Consider groups of the form $\operatorname{FSym}(\mathbf N) \times G$, where $G$ is any countable group. Obviously any such group contains a copy of every finite group. I am not sure whether $\op …

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 …

YCor beat me to it, but I will post my answer anyway because it is rather different. I will construct a counterexample to the first question with $\Gamma = F_2 = F\{x,y\}$, the free group on two gener …

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 …

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 …

Recall that $F_2$ is $\pi_1(\infty)$, where $\infty =S^1 \vee S^1$ is a bouquet of two circles. Let $H \le F_2$ be a subgroup. Then $H \cong \pi_1(\Gamma, v)$ for some covering space $\Gamma$ of $S^1\ …

This is an answer to the follow-up question about automorphisms of a subdivision. Suppose $G$ is a connected graph which is not $2$-regular. Let $G^{(k)}$ be the $k$-subdivision of $G$, i.e., the gra …

Converting my comment into an answer: Let $G = G_1 \ge G_2 \ge \cdots$ be the lower central series. Then $G_k/G_{k+1}$ is spanned by the left-normed commutators $[x_1, \dotsc, x_k]$ with $x_1, \dotsc, …

Let $G_1$ and $G_2$ be nonisomorphic Sylow-isomorphic groups. For example let $G_1 = C_6$ and $G_2 = S_3$. Then for any finite group $H$, the groups $G_1 \times H$ and $G_2 \times H$ are nonisomorphic …

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 …