Groups with some arithmetic conditions on real class sizes

Abstract

Let G be a finite group. An element x in G is a real element if x is conjugate to its inverse in G. For x in G, the conjugacy class x^G is said to be a real conjugacy class if every element of x^G is real. We show that if 4 divides no real conjugacy class size of a finite group G, then G is solvable. We also study the structure of such groups in detail. This generalizes several results in the literature.Comment: 9 page

Invariant class sizes and solvability of finite groups under coprime action

“…For our purposes, we need some properties regarding ordinary class sizes, some of which are known, but not all. The following result, Lemma 2.4, was originally proved in [4] in the particular case of considering the class sizes of all elements of G. Recently, a similar result has been published with a restrictive additional hypothesis (taking into account real odd prime power order elements) in Theorem A of [11] by using the CFSG. Therefore, Lemma 2.4 is a trivial consequence of it.…”

The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph

“…Recognizing p-nilpotency (i.e., has a normal p-complement or not) is clearly an important problem. In this section, we provide some evidence for a conjecture proposed in [8], which said that, if the lengths of all the nontrivial real classes in G have the same 2-part, then O 2 (G) is 2-nilpotent; in particular, G is solvable. In Theorem 4, we prove that, with only real classes of primary elements being considered, this conjecture is held under some condition on the Sylow 2-subgroups.…”

“…In particular, the problems of recognizing the solvability of G and the existence of normal (or abelian) p-complements of G in this context have been widely investigated, which are also the major themes of our discussion here. Recently, some authors do their research by placing conditions on only real classes of G. Fruitful achievements are obtained in this line of study, and it turns out that the lengths of these classes are closely related to some fundamental properties of G, see, for instance, [2,7,8]. The new techniques developed in those papers are valuable and some analogue methods are employed in this article.…”

“…Now, we mention several notable findings related to real classes. In ( [8], Theorem 4.2), Tong-Viet proved that, if the class length of every real primary 2 -element of G is either a 2-power or not divisible by 4, then G is solvable. It was shown in ( [2], Theorem 6.1) that 2 is not a vertex of Γ r (G) if and only if G has a normal Sylow 2-subgrpup S (i.e., G is 2-closed) and Re(S) ⊆Z(S).…”

Real class sizes

“…Using [3], we see that L/U has a self-normalizing Sylow 2-subgroup T /U and a real element Uz ∈ L/U of order 3 with |(Uz) L/U | = 7 · 8. There exists a real element y ∈ L of 3-power order with Uz = Uy (see [16,Lemma 2.6]). Since |(Uz) L/U | divides |y L | and |y L | divides |y G |, we see that 7 ∈ π 1 .…”