A generating set S for a group G is independent if the subgroup generated by $S\setminus \{s\}$ is properly contained in G for all $s \in S.$ We describe the structure of finite groups G such that there are precisely two numbers appearing as the cardinalities of independent generating sets for G.

No group has exactly one or two nonpower subgroups. We classify groups containing exactly three nonpower subgroups and show that there is a unique finite group with exactly four nonpower subgroups. Finally, we show that given any integer k greater than $4$ , there are infinitely many groups with exactly k nonpower subgroups.

Let $m\leqslant n\in \mathbb {N}$, and $G\leqslant \operatorname {Sym}(m)$ and $H\leqslant \operatorname {Sym}(n)$. In this article, we find conditions enabling embeddings between the symmetric R. Thompson groups ${V_m(G)}$ and ${V_n(H)}$. When $n\equiv 1 \mod (m-1)$, and under some other technical conditions, we find an embedding of ${V_n(H)}$ into ${V_m(G)}$ via topological conjugation. With the same modular condition, we also generalize a purely algebraic construction of Birget from 2019 to find a group $H\leqslant \operatorname {Sym}(n)$ and an embedding of ${V_m(G)}$ into ${V_n(H)}$.

Morgan and Parker proved that if G is a group with ${\textbf{Z}(G)} = 1$ , then the connected components of the commuting graph of G have diameter at most $10$ . Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a $2$ -Frobenius group, and if the commuting graph of G is connected, then its diameter is at most $8$ . We prove that the hypothesis $Z (G) = 1$ in these results can be replaced with $G' \cap {\textbf{Z}(G)} = 1$ . We also prove that if G is solvable and $G/{\textbf{Z}(G)}$ is either a Frobenius group or a $2$ -Frobenius group, then the commuting graph of G is disconnected.

Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,_{n}x]$ over $x\in G$. In the case when $G$ is soluble we prove that if, for some $n$, the Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$, it is proved that if, for some $n$, the generalized Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the generalized Fitting subgroup $F_{f(k,m)}^{\ast }(G)$ with $f(k,m)$ depending only on $k$ and $m$, where $|g|$ is the product of $m$ primes counting multiplicities. It is also proved that if, for some $n$, the nonsoluble length of $R_{n}(g)$ is equal to $k$, then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$. Earlier, similar generalizations of Baer’s theorem (which states that an Engel element of a finite group belongs to the Fitting subgroup) were obtained by the first two authors in terms of left Engel-type subgroups.

In this paper, we introduce the notion of the equivalence relation, called n-isoclinism, between crossed modules of groups, and give some basic properties of this notion. In particular, we obtain some criteria under which crossed modules are n-isoclinic. Also, we present the notion of n-stem crossed module and, under some conditions, determine them within an n-isoclinism class.

In this paper we prove that every group with at most 26 normalisers is soluble. This gives a positive answer to Conjecture 3.6 in the author’s paper [On groups with a finite number of normalisers’, Bull. Aust. Math. Soc.86 (2012), 416–423].

We show that there are only finitely many primes $p$ such that $\mathrm{PSL} (2, p)$ has a minimal generating set of size four.

Supplementary materials are available with this article.

Let α be a formation of finite groups which is closed under subgroups and group extensions and which contains the formation of solvable groups. Let G be any finite group. We state and prove equivalences between conditions on chief factors of G and structural characterizations of the α-residual and theα-radical of G. We also discuss the connection of our results to the generalized Fitting subgroup of G.

Let (M, G) be a pair of groups, in which M is a normal subgroup of G such that G/M and M/Z(M, G) are of orders pm and pn. respectively. In 1998, Ellis proved that the commutator subgroup [M, G] has order at most pn(n + 2 m−1)/2.

In the present paper by assuming /[M, G] = pn(n+2m−1)/2, we determine the pair (M, G). An upper bound is obtained for the Schur multiplier of the pair (M, G), which generalizes the work of Green (1956).

R. Gow proved that the order of a solvable rational group is divisible only by the primes 2, 3 and 5. In this paper it is proved that in a solvable rational group the Sylow 5-subgroup is always normal and elementary Abelian. Moreover, the structure of rational {2, 5}-groups is described in detail.

In this paper we prove the following:

1. 1. Let $m\ge 2,\,n\ge 1$ be integers and let $G$ be a group such that ${{(XY)}^{n}}\,=\,{{(YX)}^{n}}$ for all subsets $X,Y$ of size $m$ in $G$. Then

   1. a) $G$ is abelian or a $\text{BFC}$-group of finite exponent bounded by a function of $m$ and $n$.

   2. b) If $m\ge n$ then $G$ is abelian or $|G|$ is bounded by a function of $m$ and $n$.

2. 2. The only non-abelian group $G$ such that ${{(XY)}^{2}}\,=\,{{(YX)}^{2}}$ for all subsets $X,Y$ of size 2 in $G$ is the quaternion group of order 8.

3. 3. Let $m$, $n$ be positive integers and $G$ a group such that

   $${{X}_{1}}\cdot \cdot \cdot \,{{X}_{n}}\,\subseteq \,\bigcup\limits_{\sigma \in {{S}_{n}}\,\backslash \,1}{{{X}_{\sigma (1)}}\cdot \cdot \cdot \,{{X}_{\sigma (n)}}}$$

for all subsets ${{X}_{i}}$ of size $m$ in $G$. Then $G$ is $n$-permutable or $|G|$ is bounded by a function of $m$ and $n$.

A group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.