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On the number of solutions of equation x p k = 1 in a finite group

Yakov Berkovich (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Theorem A yields the condition under which the number of solutions of equation x p k = 1 in a finite p -group is divisible by p n + k (here n is a fixed positive integer). Theorem B which is due to Avinoam Mann generalizes the counting part of the Sylow Theorem. We show in Theorems C and D that congruences for the number of cyclic subgroups of order p k which are true for abelian groups hold for more general finite groups (for example for groups with abelian Sylow p -subgroups).

On the recognizability of some projective general linear groups by the prime graph

Masoumeh Sajjadi (2022)

Commentationes Mathematicae Universitatis Carolinae

Let G be a finite group. The prime graph of G is a simple graph Γ ( G ) whose vertex set is π ( G ) and two distinct vertices p and q are joined by an edge if and only if G has an element of order p q . A group G is called k -recognizable by prime graph if there exist exactly k nonisomorphic groups H satisfying the condition Γ ( G ) = Γ ( H ) . A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that PGL ( 2 , p α ) is recognizable, if p is an odd prime and α > 1 is odd. But for even α , only the recognizability...

On zeros of characters of finite groups

Jinshan Zhang, Zhencai Shen, Dandan Liu (2010)

Czechoslovak Mathematical Journal

For a finite group G and a non-linear irreducible complex character χ of G write υ ( χ ) = { g ∈ G ∣ χ ( g ) = 0 } . In this paper, we study the finite non-solvable groups G such that υ ( χ ) consists of at most two conjugacy classes for all but one of the non-linear irreducible characters χ of G . In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable ϕ -groups. As a corollary, we answer Research Problem 2 in [Y. Berkovich and L. Kazarin: Finite...

Permutability of centre-by-finite groups

Brunetto Piochi (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let G be a group and m be an integer greater than or equal to 2 . G is said to be m -permutable if every product of m elements can be reordered at least in one way. We prove that, if G has a centre of finite index z , then G is ( 1 + [ z / 2 ] ) -permutable. More bounds are given on the least m such that G is m -permutable.

Prime divisors of conjugacy class lengths in finite groups

Carlo Casolo (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We show that in a finite group G which is p -nilpotent for at most one prime dividing its order, there exists an element whose conjugacy class length is divisible by more than half of the primes dividing G / Z ⁢ G .

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