Classification of EPPO-groups with seven non-central conjugacy classes

Subject Areas : Statistics

Zeinab  Foruzanfar ¹ (Department of Mathematics Imam Khomeini International University - Buin Zahra Higher Education Center of Engineering and Technology, Qazvin, Iran)
Mehdi  Rezaei ² (Department of Mathematics Imam Khomeini International University - Buin Zahra Higher Education Center of Engineering and Technology, Qazvin, Iran)

Received: 2020-12-06 Accepted : 2021-10-02 Published : 2022-06-22

Keywords: گروه فروبنیوس, p -زیرگروه سیلو, مرتبه, کلاس تزویج,

Abstract :

Let G be a finite group and Z(G) be a subgroup of it. Suppose that for the finite group G, Pi_e(G) denotes the set of orders of elements of G. Then G is an EPPO-group if the orders of its elements are non-negative powers of primes. Also, for a subset A of G, let r_G(A) be the number of conjugacy classes of G that intersect A non-trivially. The purpose of this paper is to classify all finite EPPO-groups with the property r_G(G-Z(G))=7. We first verify the case where Z(G)=1. Then we verify the case where G/Z(G) is abelian. After that, we consider the case where G/Z(G) is non-abelian. We verify this case in three subcases where G/Z(G) is a p-group, a Frobenius group or a 2-Frobenius group. In fact, we show that the only groups which satisfy the intended property are the groups that are attained in the case Z(G)=1 and all of these groups are Frobenius groups.

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