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Let $G$ be a $d$-generated group. Then my first question is how to see free reduced group $FV(G)$ in the variety containing $G$. What I understood is: "Let $W \subset F_d$ ($d$-generated free group) b …

Let $G$ be a finite supersolvable group, and if $p$ is the biggest prime dividing $\vert G \vert$. Then $G$ has normal subgroups of order every possible power of $p$. Analogous statement in case of pr …

Dan Segal, in his book 'Words', has defined generalized words. I have trouble understanding generalized words. What I have understood from the definition of generalized words are as follows: Let $X = …

Let $G$ be a group. A subgroup $H$ of $G$ is said to be fully invariant if for every endomorphism $\phi $ of $G$, we have $\phi(H) \subseteq H$. For a finitely generated residually finite group $G$, l …

Let $w$ be a word in free group $F$ on finitely many generators. We will look at $w$ as word map on groups. It is clear that there exists an endomorphism $\phi$ of $F$ such that $\phi(w) = w^{-1}$ if …

Let $w$ be a word in a free group $F_2$ of two generators $x_1, x_2$ such that there does not exist any endomorphism of free group which takes $w$ to $w^{-1}$. Let $w_1, w_2$ be two words in the same …

Let $F_n$ be a free group on $n$ generators. Let $w \in F_n$ be a word such that there does not exist any solution in $F_n$ for the equation $w.w(t_1, \ldots, t_n) = 1$, where $t_1, \ldots, t_n$ are v …

Let $G$ be a group. An element $g \in G$ is said to be a test element if any endomorphism $\phi$ of $G$ such that $\phi(g) = g$ is an automorphism. The free group $F_2$ of rank $2$ is generated by $x_ …