A finite group is said to be n-cyclic if it contains n cyclic subgroups. For a finite group G, the ratio of the number of cyclic subgroups to the number of subgroups is known as the cyclicity degree of the group G and is denoted by $\textit {cdeg} (G)$. In this paper, we classify all $12$-cyclic groups. We also prove that the set of cyclicity degrees for all the finite groups is dense in $[0,1]$, which solves a problem posed by Tărnăuceanu and Tóth [‘Cyclicity degrees of finite groups’, Acta Math. Hungar. 145(2) (2015), 489–504].

In 1954, B. H. Neumann discovered that if $G$ is a group in which all conjugacy classes have finite cardinality at most $m$, then the derived group $G'$ is finite of $m$-bounded order. In 2018, G. Dierings and P. Shumyatsky showed that if $|x^G| \le m$ for any commutator $x\in G$, then the second derived group $G''$ is finite and has $m$-bounded order. This paper deals with finite groups in which $|x^G|\le m$ whenever $x\in G$ is a commutator of prime power order. The main result is that $G''$ has $m$-bounded order.

We strengthen two results of Moretó. We prove that the index of the Fitting subgroup is bounded in terms of the degrees of the irreducible monomial Brauer characters of the finite solvable group G and it is also bounded in terms of the average degree of the irreducible Brauer characters of G that lie over a linear character of the Fitting subgroup.

For a group G and $m\ge 1$, let $G^m$ denote the subgroup generated by the elements $g^m$, where g runs through G. The subgroups not of the form $G^m$ are the nonpower subgroups of G. We classify the groups with at most nine nonpower subgroups.

We prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G, then the number of conjugacy classes of G is at least $Dp/\log_2p$. We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$.

Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p-groups.

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.

For any given subgroup H of a finite group G, the Quillen poset ${\mathcal {A}}_p(G)$ of nontrivial elementary abelian p-subgroups is obtained from ${\mathcal {A}}_p(H)$ by attaching elements via their centralisers in H. We exploit this idea to study Quillen’s conjecture, which asserts that if ${\mathcal {A}}_p(G)$ is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the ${{\mathbb {Z}}}$-acyclic version of the conjecture (obtained by replacing ‘contractible’ by ‘${{\mathbb {Z}}}$-acyclic’). We also work with the ${\mathbb {Q}}$-acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least $2$. This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most $4$.

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.

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.

Let H be a subgroup of a group G. Then, we call H weakly s-supplemented in G if G has a subgroup T such that HT = G and H ∩ T ≤ HsG, where HsG is the largest s-permutable subgroup of G contained in H. In this paper, we use the weakly s-supplemented subgroups to characterize the structure of groups. A series of known results in the literature are unified and generalized.

A complete Sylow sequence, 𝒫=P1,…,Pm, of a finite group G is a sequence of m Sylow pi-subgroups of G, one for each pi, where p1,…,pm are all of the distinct prime divisors of |G|. A product of the form P1⋯Pm is called a complete Sylow product of G. We prove that the solvable radical of G equals the intersection of all complete Sylow products of G if, for every composition factor S of G, and for every ordering of the prime divisors of |S|, there exist a complete Sylow sequence 𝒫 of S, and g∈S such that g is uniquely factorizable in 𝒫 . This generalizes our results in Kaplan and Levy [‘The solvable radical of Sylow factorizable groups’, Arch. Math.85(6) (2005), 490–496].

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.

In this paper we describe the groups admitting a covering with Hall subgroups. We also determine the groups with a π1-Hall subgroup, where π1 is the connected component of the prime graph, containing the prime 2.

It is shown that a Carter subgroup of a non-soluble unitary group can only be the normalizer of a Sylow 2-subgroup. This result, combined with previous results, implies that no finite simple group can be a minimal counterexample to the conjugacy conjecture of Carter subgroups in a finite group.The third author was partially supported by the University of Padova, the RFBR, grants 02-01-00495 and 02-01-06226, Ministry of Education RF, E00-1.0-77, ‘Universities of Russia’ UR.04.01.031.

Coverings of certain simplical complexes constructed from subgroups of group G are related to covering groups of G.

We obtain the following generalization of a well known result of Huppert. If p is the largest primer divisor of the order of a finite group G and q is any prime distinct from p, then G is supersolvable if and only if every maximal subgroup whose index is relatively prime to either p or q, has prime index.