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The answer to the first part of OP's question is "no", but possibly with a finite number of exceptions. Indeed [Theorem 1.8, 4] established that the number of $T_2$-systems of a non-abelian finite s …

The goal of my answer is to provide references to articles that studied the class of groups under consideration. Let $\mathcal{C}$ be the class of the finitely generated groups in which every weakly …

The solvable Baumslag-Solitar group $BS(1, n) = \langle a, b \, \vert \, aba^{-1} = b^n \rangle$ with $n \in \mathbb{Z} \setminus \{0\}$, has only one $T$-system of generating pairs and any number of …

In my answer to your other question, I give references supporting the fact that almost all finite non-abelian simple groups aren't "almost nice". So I would rather ask: Is there an handy characterizat …

This is a complement to Johannes Hahn's answer. Corrigendum. In the previous version of this answer, I have made an erroneous claim, allowing $\omega$, the order of $A$ and $B$, to be any positive num …

As a complement to YCor's elegant answer, I would like to present three additional ways to prove YCor's Statement 1. The groups $G_m$ and $G_n$ are isomorphic if and only if $m = n$. The …

The group law of $G^{\rtimes 2}$ is given by $$(g, h)(g',h') = (ghg'h^{-1}, hh')$$ and more generally for $G^{\rtimes k}$, we have $(g_1, \dots, g_{k -1}, g_k)^{(1, \dots,1, g)} = (g_1^g, \dots,g_{k …

The answer is no. In order to see this, you may combine Theorem 3.9 and Remark 5.3 of [1]. A counter-example is given by the nilpotent-by-Abelian group $B$ of Equation (3.2). Further examples are p …

The Jacobson radical $\mathfrak{J}(G)$ of a group $G$ is defined by Reinhold Baer in [2] as the intersection of the maximal normal subgroups of $G$ (as noted by Baer the identity $G = \mathfrak{J}(G)$ …

Let us call $G$ a generalised Dedekind group if every subgroup of $G$ is isomorphic to a normal subgroup of $G$. As expected, Dedekind groups are generalised Dedekind groups. In addition, YCor has est …

Claim. The groups $G_A$ and $G_B$ are not isomorphic. We will use the following lemmas. Lemma 1. Let $A \in \text{GL}_n(\mathbb{Z})$ and let $G_A \Doteq \mathbb{Z} \ltimes_A \mathbb{Z}^n$. Then the …

This is only a long comment. Claim. If $G$ is a $2$-generated finite group such that $(x, yxy^{-1})$ generates $G$ whenever $(x, y)$ does, then $G$ is perfect. Proof. Following this …

Here is my attempt to entertain those who knew the answer. Question 1. Let $G$ be a finitely generated group such that $G/\Phi(G)$ is Abelian. Is $G$ nilpotent? The answer is yes if we a …

Some fairly general lower bounds are given in Igor Pak's "What do we know about the product placement algorithm" [2]. Let $d(G)$ denote the minimal number of generators of $G$. By $\varphi_{k}(G)$ we …

Here is a proof which relies on a straighforward generalization of the Ping Pong Lemma. Claim 1 Let $a$ and $b$ be the transformations of the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C} \cup \{\inf …