The Chermak–Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we prove that, for any finite abelian group A, there exists a finite group G such that the Chermak–Delgado lattice of G is a subgroup lattice of A.

It is shown that, for every prime number p, the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gp of all finite p-groups has the cardinality of the continuum. Furthermore, it is shown, in addition, that the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gsol of all finite solvable groups has also the cardinality of the continuum.

Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathscr{P}$ be the class of $p$-groups of order $p^{n}$ ($n\geq 3$). Consider the function $\unicode[STIX]{x1D6FC}:\mathscr{P}\longrightarrow (0,1]$ given by $\unicode[STIX]{x1D6FC}(G)=|C(G)|/|G|$. In this paper, we determine the second minimum value of $\unicode[STIX]{x1D6FC}$, as well as the corresponding minimum points. Since the problem of finding the second maximum value of $\unicode[STIX]{x1D6FC}$ has been solved for $p=2$, we focus on the case of odd primes in determining the second maximum.

A subset $X$ of a finite group $G$ is a set of pairwise noncommuting elements if $xy\neq yx$ for all $x\neq y\in X$. If $|X|\geq |Y|$ for any other subset $Y$ of pairwise noncommuting elements, then $X$ is called a maximal subset of pairwise noncommuting elements and the size of such a set is denoted by ${\it\omega}(G)$. In a recent article by Azad et al. [‘Maximal subsets of pairwise noncommuting elements of some finite $p$-groups’, Bull. Iran. Math. Soc.39(1) (2013), 187–192], the value of ${\it\omega}(G)$ is computed for certain $p$-groups $G$. In the present paper, our aim is to generalise these results and find ${\it\omega}(G)$ for some more $p$-groups of interest.

The subgroup commutativity degree of a group $G$ is the probability that two subgroups of $G$ commute, or equivalently that the product of two subgroups is again a subgroup. For the dihedral, quasi-dihedral and generalised quaternion groups (all of 2-power cardinality), the subgroup commutativity degree tends to 0 as the size of the group tends to infinity. This also holds for the family of projective special linear groups over fields of even characteristic and for the family of the simple Suzuki groups. In this short note, we show that the family of finite $P$-groups also has this property.

We prove a cohomological property for a class of finite $p$-groups introduced earlier by Xu, which we call semi-abelian $p$-groups. This result implies that a semi-abelian $p$-group has noninner automorphisms of order $p$, which settles a long-standing problem for this class. We answer also, independetly, an old question posed by Xu about the power structure of semi-abelian $p$-groups.

A long-standing conjecture asserts that every finite nonabelian $p$-group has a noninner automorphism of order $p$. In this paper the verification of the conjecture is reduced to the case of $p$-groups $G$ satisfying ${ Z}_{2}^{\star } (G)\leq {C}_{G} ({ Z}_{2}^{\star } (G))= \Phi (G)$, where ${ Z}_{2}^{\star } (G)$ is the preimage of ${\Omega }_{1} ({Z}_{2} (G)/ Z(G))$ in $G$. This improves Deaconescu and Silberberg’s reduction of the conjecture: if ${C}_{G} (Z(\Phi (G)))\not = \Phi (G)$, then $G$ has a noninner automorphism of order $p$ leaving the Frattini subgroup of $G$ elementwise fixed [‘Noninner automorphisms of order $p$ of finite $p$-groups’, J. Algebra 250 (2002), 283–287].

In this paper we prove that every nonabelian finite 2-group with a cyclic commutator subgroup has a noninner automorphism of order two fixing either Φ(G) or Z(G) elementwise. This, together with a result of Peter Schmid on regular p-groups, extends our result to the class of nonabelian finite p-groups with a cyclic commutator subgroup.

We consider classifying spaces of a family of p-groups and prove that mod p cohomology enriched with Bockstein spectral sequences determines their homotopy type among p-completed CW-complexes.

Let p be an odd prime and let G be a finite p-group such that xZ(G)⊆xG, for all x∈G∖Z(G), where xG denotes the conjugacy class of x in G. Then G has a noninner automorphism of order p leaving the Frattini subgroup Φ(G) elementwise fixed.

We consider finite p-groups G in which every cyclic subgroup has at most p conjugates. We show that the derived subgroup of such a group has order at most p2. Further, if the stronger condition holds that all subgroups have at most p conjugates then the central factor group has order p4 at most.