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In this paper, we investigate the structure of finite solvable groups G admitting a normal subgroup N such that G / N {G/N} is non-abelian and N contains all but four conjugacy classes of G.

For a finite group 𝐺, an element is called a vanishing element of 𝐺 if it is a zero of an irreducible character of 𝐺; otherwise, it is called a non-vanishing element. Moreover, the conjugacy class of an element is called a vanishing class if that element is a vanishing element. In this paper, we describe finite groups whose vanishing class sizes are all prime powers, and on the other hand we show that non-vanishing elements of such a group lie in the Fitting subgroup which is a proof of a conjecture mentioned in [I. M. Isaacs, G. Navarro and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra 222 (1999), 2, 413–423] under this special restriction on vanishing class sizes.

Let G be a finite group. A classical theorem of Burnside shows that a group with order divisible by at most two primes is solvable. The aim of this article is studying finite groups each of whose v...

Let m > 1 be a fixed positive integer. In this paper, we consider finite groups each of whose nonlinear character degrees has exactly m prime divisors. We show that such groups are solvable whenever m > 2. Moreover, we prove that if G is a non-solvable group with this property, then m = 2 and G is an extension of A7 or S7 by a solvable group.

Let G be a finite group and let π(G) be the set of primes dividing the order of G. For each p∈π(G), the Sylow theorems state that the number of Sylow p-subgroups of G is equal to kp + 1 for some no...

Let G be a group, and let [Formula: see text](G) be a digraph whose vertices are the non-trivial conjugacy classes of G and there is an arc from a vertex C to a vertex C′ if and only if C ≠ C′ and 〈C〉 ⊆ 〈C′〉. In this paper, we characterize finite groups G whose associated digraphs [Formula: see text](G) are oriented trees.

Let G be a finite group. I. M. Isaacs and G. Seitz have conjectured that if G is a solvable group, then Taketa’s inequality dl(G)≤|cd(G)| holds, where cd(G) is the set of irreducible charac...

Let [Formula: see text] be a finite group. We say that a conjugacy class of [Formula: see text] in [Formula: see text] is vanishing if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

Let G be a finite group. Isaacs and Seitz have conjectured that an upper bound for the derived length dl(G) of G is the number of irreducible character degrees |cd(G)| of G, whenever G is solvable....

Let $G$ be a finite group. We say that an element $g$ in $G$ is a vanishing element if there exists some irreducible character $\chi$ of $G$ such that $\chi(g)=0$. In this paper, we prove that if the set of vanishing elements of $G$ is the union of at most three conjugacy classes, then $G$ is solvable.

In this paper, we determine all almost simple groups each of whose character degrees has at most two distinct prime divisors. More generally, we show that a finite non-solvable group [Formula: see text] with this property is an extension of an almost simple group [Formula: see text] by a solvable group and [Formula: see text], where [Formula: see text] is the set of all primes dividing some character degree of [Formula: see text].

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.

Let G a finite group. For conjugacy classes A and B of G, the subset AB of G is a union of conjugacy classes of G, and for irreducible complex characters χ and ϕ of G, the product χϕ is a linear combination of irreducible characters of G with positive integer coefficients. In this article we give the structure of finite groups G whenever C 2 contains a central conjugacy class or does not contain a central conjugacy class of G, for all conjugacy classes C of G. Also in the case of (C −1) m C n ⊆ Z(G) the structure of G is given. Similar results when C is replaced by an irreducible character χ of G are discussed.

Let [Formula: see text] be a finite group. We say that an element [Formula: see text] in [Formula: see text] is a vanishing element if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we study the structure of finite groups whose vanishing elements are of odd order.

Let G be a finite group. If A and B are two conjugacy classes in G, then AB is a union of conjugacy classes in G and h(AB) denotes the number of distinct conjugacy classes of G contained in AB. If c and y are two complex irreducible characters of G, then cy is a character of G and again we let h(cy) be the number of irreducible characters of G appearing as constituents of cy. In this paper our aim is to study the product of conjugacy classes in a finite group and obtain an upper bound for h in general. Then we study similar results related to the product of two irreducible characters.

Let G be a finite group. We define the derived covering number and the derived character covering number of G, denoted respectively by dcn (G) and dccn (G), as the smallest positive integer n such that Cn = G′ for all non-central conjugacy classes C of G and Irr ((χn)G′) = Irr (G′) for all nonlinear irreducible characters χ of G, respectively. In this paper, we obtain some results on dcn and dccn for a finite group G, such as the existence of these numbers and upper bounds on them.