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The following nice argument, due to Alexander Olshanskii, shows that the answer to both questions is negative. It uses the theorem of Golod-Shafarevich (see http://arxiv.org/abs/1206.0490 for a surve …

I think that Higman's group H has plenty of such normal subgroups. Indeed, let G be the extension of H with the automorphism h. Then H has index 4 in G. By Schupp's theorem, H is SQ-universal, hence t …

Let $F$ be a finitely generated non-abelian free group with a non-trivial normal subgroup $R \lhd F$. Suppose that $G:=F/R$ is finitely presented. Then the group $F/R'$ is finitely presented if and on …

Well, any finitely generated group G of exponent 3 is finite by a classical theorem of Burnside. And since the order of every element is 3, the order of G must be a power of 3 by Cauchy's theorem. It …

A modification of Guntram's example could produce a countable group with the required property, which is not an FC-group. Let $G$ be the direct product of non-abelian symmetric groups $G=\times_{n\ge …

Yes, such groups exist. Consider the disjoint union of cyclic groups of odd order $\mathcal{C}:=\{ \mathbb{Z}/(2n+1)\mathbb{Z} \mid n \in \mathbb{N} \}$. By a theorem of A. Ol'shanskii (see Thm 35. …

The group $G$ maps onto the free product $C_p*C_p*C_p$ of three cyclic groups of order $p$ (just send each $b_i^p$ to $1$). This free product is virtually free, as a free product of finite groups (by …

Here is a question that has been bothering me for some time: Let G be a finitely generated conjugacy separable group with solvable word problem. Does it follow that the conjugacy problem in G is solv …

Let $F=F(a,b)$ be the free group of rank $2$. Question 1: Given any positive integer $d$, can one always find elements $u_j,v_j,w_j \in F$, $j=1,\dots,d$, such that if $1 \le j <k \le d$ then the no …

A well-known theorem of Gruenberg implies that a finitely generated residually torsion-free nilpotent group is residually $p$-finite for all primes $p$. What about the converse? Question: Are there a …

Browsing through the archive of solved problems of Kourovka Notebook, I accidentally saw that the same question was asked by Yu.V. Kuz'min in 1999 (see question 14.52). Apparently the required exampl …

It's not hard to construct counter-examples for large exponents. Consider the following two automorphisms $\xi, \eta$ of the free group $F=F(x,y)$ of rank $2$, defined by $$\xi(x)=y,~\xi(y)=x \text{ a …

Call a group presentation $\langle X \,\|\,R \rangle$ minimal if no relator from $R$ is a consequence of the remaining relators, i.e., no $r \in R$ belongs to the normal closure of $R\setminus \{r\}$ …

Let $G$ be a finitely generated virtually abelian group (i.e., $G$ contains $\mathbb{Z}^n$ with finite index for some $n\ge 2$). Is there anything known about the outer automorphism group $Out(G)$? …

A. Yu. Olshanskii in the paper "Periodic quotient groups of hyperbolic groups." ((Russian) Mat. Sb. 182 (1991), no. 4, 543--567; translation in Math. USSR-Sb. 72 (1992), no. 2, 519–541) proved that fo …