<Emphasis Type="Italic">X</Emphasis>-decomposable finite groups for <Emphasis Type="Italic">X</Emphasis>={1, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">m</Emphasis> + 1, <Emphasis Type="Italic">m</Emphasis> + 2}

A normal subgroup N of a finite group G is called n-decomposable in G if N is the union of n distinct G-conjugacy classes. We study the structure of nonperfect groups in which every proper nontrivial normal subgroup is m-decomposable, m+1-decomposable, or m+2-decomposable for some positive integer m. Furthermore, we give classification for the soluble case.     

Triangles in the graph of conjugacy classes of normal subgroups

Let G be a finite group and N a normal subgroup of G. We determine the structure of N when the graph \(\Gamma _{G}(N)\), which is the graph associated to the conjugacy classes of G contained in N, has no triangles and when the graph consists in exactly one triangle.     

Multiplying a conjugacy class by its inverse in a finite group

Suppose that G is a finite group and K is a non-trivial conjugacy class of G such that KK^?1 = 1 ∪ D ∪ D^?1 with D a conjugacy class of G. We prove that G is not a non-abelian simple group and we give arithmetical conditions on the class sizes determining the solvability and the structure of 〈K〉 and 〈D〉.     

Equalities on the number of conjugacy classes of the Sylow p-subgroups of GL(n,q)

We denote by Gn the group of the upper unitriangular matrices over F_q, the finite field with q = p^t elements, and r(G_n) the number of conjugacy classes of G_n. In this paper, we obtain the value of r(G_n) modulo (q² -1)(q -1). We prove the following equalities     

Finite groups with some non-Abelian subgroups of non-prime-power order

Let G be a finite group and NA(G) denote the number of conjugacy classes of all nonabelian subgroups of non-prime-power order of G. The Symbol π(G) denote the set of the prime divisors of |G|. In this paper we establish lower bounds on NA(G). In fact, we show that if G is a finite solvable group, then NA(G) = 0 or NA(G) ≥ 2^|π(G)|?2, and if G is non-solvable, then NA(G) ≥ |π(G)| + 1. Both lower bounds are best possible.     

On Thompson’s conjecture for alternating and symmetric groups of degree greater than 1361

Let G be a finite group G, and let N(G) be the set of sizes of its conjugacy classes. It is shown that, if N(G) equals N(Alt _n ) or N(Sym _n ), where n > 1361, then G has a composition factor isomorphic to an alternating group Altm with m ≤ n and the interval (m, n] contains no primes.     

Finite <Emphasis Type="Italic">p</Emphasis>-groups with a class of complemented normal subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.     

Intersections of Primary Subgroups in Nonsoluble Finite Groups Isomorphic to <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>)

Given a finite group G with socle isomorphic to L _n (2 ^m ), we describe (up to conjugacy) all ordered pairs of primary subgroups A and B in G such that A ∩ B ^g ≠ 1 for all g ∈ g.     

The power graph on the conjugacy classes of a finite group

A digraph \({\overrightarrow{\mathcal{Pc}}(G)}\) is said to be the directed power graph on the conjugacy classes of a group G, if its vertices are the non-trivial conjugacy classes of G, and there is an arc from vertex C to C′ if and only if \({C \neq C'}\) and \({C \subseteqq {C'}^{m}}\) for some positive integer \({m > 0}\). Moreover, the simple graph \({\mathcal{Pc}(G)}\) is said to be the (undirected) power graph on the conjugacy classes of a group G if its vertices are the conjugacy classes of G and two distinct vertices C and C′ are adjacent in \({\mathcal{Pc}(G)}\) if one is a subset of a power of the other. In this paper, we find some connections between algebraic properties of some groups and properties of the associated graph.     

Variations of Landau’s theorem for p-regular and p-singular conjugacy classes

The well-known Landau’s theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with O_p(G) = 1 then |G/F(G)|_p' is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and kp-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).     

Conjugacy classes of derangements in finite transitive groups

Let G be a permutation group acting transitively on a finite set Ω. We classify all such (G, Ω) when G contains a single conjugacy class of derangements. This was done under the assumption that G acts primitively by Burness and Tong-Viet. It turns out that there are no imprimitive examples. We also discuss some results on the proportion of conjugacy classes which consist of derangements.     

Intersections of two nilpotent subgroups in finite groups with socle <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Given a finite group G with socle isomorphic to L ₂(q), q ≥ 4, we describe, up to conjugacy, all pairs of nilpotent subgroups A and B of G such that A ∩ B ^g ≠ 1 for all g ∈ G.     

Normal coverings of solvable groups

For a finite non cyclic group G, let γ(G) be the smallest integer k such that G contains k proper subgroups H ₁, . . . , H _k with the property that every element of G is contained in \({H_i^g}\) for some \({i \in \{1,\dots,k\}}\) and \({g \in G.}\) We prove that for every n ≥ 2, there exists a finite solvable group G with γ(G) = n.     

On semi-rational finite simple groups

An element of a group G is called semi-rational if all generators of \(\langle x\rangle \) lie in the union of two conjugacy classes of G. If all elements of G are semi-rational, then G is called a semi-rational group. In this paper, we determine all semi-rational simple groups. Our study in this article generalises Feit and Seitz’s result (Ill J Math 33(1):103–131, 1989) to semi-rational groups.     

Finite groups with three rational conjugacy classes

We show that if a finite group G has exactly three rational conjugacy classes, then G also has exactly three rational-valued irreducible complex characters. This generalizes a result of Navarro and Tiep (Trans Amer Math Soc 360:2443–2465, 2008) and partially answers in the affirmative a conjecture of theirs. We also give a family of examples of non-solvable groups with exactly three rational conjugacy classes.     

Multiplication of conjugacy classes,colligations, and characteristic functions of matrix argument

We extend the classical construction of operator colligations and characteristic functions. Consider the group G of finitary block unitary matrices of order α+∞+···+∞ (m times) and its subgroup K ? U(∞), which consists of block diagonal unitary matrices with the identity block of order α and a matrix u ∈ U(∞) repeated m times. It turns out that there is a natural multiplication on the space G//K of conjugacy classes. We construct “spectral data” of conjugacy classes, which visualize the multiplication and are sufficient for reconstructing a conjugacy class.     

On finite groups for which the lattice of <Emphasis Type="Italic">S</Emphasis>-permutable subgroups is distributive

Let p be a prime and let P be a Sylow p-subgroup of a finite nonabelian group G. Let bcl(G) be the size of the largest conjugacy classes of the group G. We show that if p is an odd prime but not a Mersenne prime or if P does not involve a section isomorphic to the wreath product \({Z_p \wr Z_p}\), then \({|P/O_p(G)| \leq bcl(G)}\).     

Subgroup normality degrees of finite groups I

In this paper, we introduce the probability that a subgroup H of a finite group G can be normal in G, the subgroup normality degree of H in G, as the ratio of the number of all pairs \({(h, g)\in H\times G}\) such that \({h^g\in H}\) by |H||G|. We give some upper and lower bounds for this probability and obtain the upper bound \({\frac{8}{15}}\) for nontrivial subgroups of finite simple groups. In addition, we obtain explicit formulas for subgroup normality degrees of some particular classes of finite groups.     

On the structure of a mutually permutable product of finite groups

Let a finite group \({G = AB}\) be the product of the mutually permutable subgroups A and B. We investigate the structure of G given by conditions on conjugacy class sizes of elements in \({A \cup B}\) . Some recent results are extended.