Finite groups with a given absolute central factor group

Abstract

We have classified, upto isoclinism, certain groups with a given central factor. As an application, we classify, upto isoclinism, groups having at the most nine element centralizers. Among other results of independent interest, we have classified, upto isoclinism, groups having a central factor of order p 3 , p a prime. All these improves some previous results. 2020 Mathematics Subject Classification. 20D60, 20D99.

The absolute center of finite groups

“…In this section we hope to extend the result of Chaboksavar, Farrokhi Derakhshandeh Ghouchan and Saeedi [3]. In fact, we determine all solutions of the equation X=L.X/ Š G whenever G Š Z p !…”

Finite Groups Which Are the Union of Autocentralizers of Some Automorphisms

“…On the contrary, one observes that [9] implies that G � Z 5 or Z 10 , which both give contradictions as Z 5 has fixed-point-free automorphism, and Z 10 does not conform to the conditions given at the beginning of the second section. erefore, (|G|/|L(G)|) ≥ 6.…”

“…Theorem 1 (see Theorem 3.1 in [9]). Let G be a group with G/L(G) � C p × C p , for any prime number p.…”

“…One can easily calculate that If (G/L(G)) � S 3 , then eorem 3.5 of [9] and example 1 (iii) imply that G � S 3 and |C S 3 (A)| � 5. Again eorem 2.2 of [9] implies that if (G/L(G)) � C 6 , then G � C 12 , which does not conform to the conditions given at the beginning of Section…”