On existence of normal p-complement of finite groups with restrictions on the conjugacy class sizes

Abstract

The greatest power of a prime $p$ dividing the natural number $n$ will be denoted by $n_p$. Let $In{d}_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite group and $p$ is a prime. We prove that if there exists an integer $\alpha >0$ such that $In{d}_G(a{)}_p\in \{1,p^{\alpha }\}$ for every $a$ of $G$ and a $p$-element $x\in G$ such that $In{d}_G(x{)}_p>1$, then $G$ includes a normal $p$-complement.

7 pages

References in corpus (3)

• On finite groups isospectral to simple classical groups
• Finite simple groups with some abelian Sylow subgroups
• On a connection between the order of a finite group and the set of conjugacy classes size

Cited by in corpus (1)

• On $p$-index extremal groups