We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p-groups with the smallest maximum size for neighbourhoods of a nontrivial element in its enhanced power graph.

Let G be a finite group and let $\chi $ be an irreducible character of G. The number $|G:\mathrm {ker}\chi |/\chi (1)$ is called the codegree of the character $\chi $. We provide several relations between the structure of G and the codegrees of the characters in a given subset of $\mathrm {Irr}(G)$, where $\mathrm {Irr}(G)$ is the set of all complex irreducible characters of G. For example, we show that if the codegrees of all strongly monolithic characters of G are odd, then G is solvable, analogous to the well-known fact that if all irreducible character degrees of a finite group are odd, then that group is solvable.

We find an upper bound for the number of groups of order n up to isomorphism in the variety ${\mathfrak {S}}={\mathfrak {A}_p}{\mathfrak {A}_q}{\mathfrak {A}_r}$, where p, q and r are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of subgroups that are maximal amongst the subgroups of the general linear group that are also in the variety $\mathfrak {A}_q\mathfrak {A}_r$.

Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G, that is, $AP=PA$. Let $A_{sG}$ be the subgroup of A generated by all S-permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S-permutable subgroups of G containing A. We prove that if G is a soluble group, then S-permutability is a transitive relation in G if and only if the nilpotent residual $G^{\mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$, that is, $G^{\mathfrak {N}}\cap A^{sG}= G^{\mathfrak {N}}\cap A_{sG}$ for every subnormal subgroup A of G.

A group $G=AB$ is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing $A \cap B$ and B permutes with every subgroup of A containing $A \cap B$. Weakly mutually permutable products were introduced by the first, second and fourth authors [‘Generalised mutually permutable products and saturated formations’, J. Algebra 595 (2022), 434–443] who showed that if $G'$ is nilpotent, A permutes with every Sylow subgroup of B and B permutes with every Sylow subgroup of A, then $G^{\mathfrak {F}}=A^{\mathfrak {F}}B^{\mathfrak {F}} $, where $ \mathfrak {F} $ is a saturated formation containing $ \mathfrak {U} $, the class of supersoluble groups. In this article we prove results on weakly mutually permutable products concerning $ \mathfrak {F} $-residuals, $ \mathfrak {F} $-projectors and $\mathfrak {F}$-normalisers. As an application of some of our arguments, we unify some results on weakly mutually $sn$-products.

Suppose that G is a finite solvable group. Let $t=n_c(G)$ denote the number of orders of nonnormal subgroups of G. We bound the derived length $dl(G)$ in terms of $n_c(G)$. If G is a finite p-group, we show that $|G'|\leq p^{2t+1}$ and $dl(G)\leq \lceil \log _2(2t+3)\rceil $. If G is a finite solvable nonnilpotent group, we prove that the sum of the powers of the prime divisors of $|G'|$ is less than t and that $dl(G)\leq \lfloor 2(t+1)/3\rfloor +1$.

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $\mathsf {S}_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for $\mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.

A subgroup A of a group G is said to be hereditarily G-permutable with a subgroup B of G, if $AB^x = B^xA$ for some element $x \in \langle A, B \rangle $. A subgroup A of a group G is said to be hereditarily G-permutable in G if A is hereditarily G-permutable with every subgroup of G. In this paper, we investigate the structure of a finite group G with all its Schmidt subgroups hereditarily G-permutable.

In this paper, we investigate finite solvable tidy groups. We prove that a solvable group with order divisible by at least two primes is tidy if all of its Hall subgroups that are divisible by only two primes are tidy.

A classical result of Baer states that a finite group G which is the product of two normal supersoluble subgroups is supersoluble if and only if Gʹ is nilpotent. In this article, we show that if G = AB is the product of supersoluble (respectively, w-supersoluble) subgroups A and B, A is normal in G and B permutes with every maximal subgroup of each Sylow subgroup of A, then G is supersoluble (respectively, w-supersoluble), provided that Gʹ is nilpotent. We also investigate products of subgroups defined above when $ A\cap B=1 $ and obtain more general results.

We describe finite soluble nonnilpotent groups in which every minimal nonnilpotent subgroup is abnormal. We also show that if G is a nonsoluble finite group in which every minimal nonnilpotent subgroup is abnormal, then G is quasisimple and $Z(G)$ is cyclic of order $|Z(G)|\in \{1, 2, 3, 4\}$ .

In this note, we investigate some products of subgroups and vanishing conjugacy class sizes of finite groups. We prove some supersolubility criteria for groups with restrictions on the vanishing conjugacy class sizes of their subgroups.

Let $\pi $ be a set of primes. We say that a group G satisfies $D_{\pi }$ if G possesses a Hall $\pi $ -subgroup H and every $\pi $ -subgroup of G is contained in a conjugate of H. We give a new $D_{\pi }$ -criterion following Wielandt’s idea, which is a generalisation of Wielandt’s and Rusakov’s results.

We study the free metabelian group $M(2,n)$ of prime power exponent n on two generators by means of invariants $M(2,n)'\to \mathbb {Z}_n$ that we construct from colorings of the squares in the integer grid $\mathbb {R} \times \mathbb {Z} \cup \mathbb {Z} \times \mathbb {R}$ . In particular, we improve bounds found by Newman for the order of $M(2,2^k)$ . We study identities in $M(2,n)$ , which give information about identities in the Burnside group $B(2,n)$ and the restricted Burnside group $R(2,n)$ .

In this paper, we study the structure of finite groups $G=AB$ which are a weakly mutually $sn$ -permutable product of the subgroups A and B, that is, A permutes with every subnormal subgroup of B containing $A \cap B$ and B permutes with every subnormal subgroup of A containing $A \cap B$ . We obtain generalisations of known results on mutually $sn$ -permutable products.

In this paper, we study the relation of the size of the class two quotients of a linear group and the size of the vector space. We answer a question raised in Keller and Yang [Class 2 quotients of solvable linear groups, J. Algebra 509 (2018), 386-396].

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with ‘solubility’ replaced by ‘nilpotence’ and ‘perfectness’, among others, are false.

These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini’s theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

For a finite group $G$, define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$, where $o(g)$ denotes the order of $g\in G$. We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$, then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.

For a character $\unicode[STIX]{x1D712}$ of a finite group $G$, the co-degree of $\unicode[STIX]{x1D712}$ is $\unicode[STIX]{x1D712}^{c}(1)=[G:\text{ker}\unicode[STIX]{x1D712}]/\unicode[STIX]{x1D712}(1)$. We study finite groups whose co-degrees of nonprincipal (complex) irreducible characters are divisible by a given prime $p$.

For a given prime $p$, we investigate the finite groups all of whose 2-minimal $p$-subgroups are complemented.