Finite Groups with Small Normal Closure of Cyclic Subgroups

Abstract

A group G is called an I(r)-group if | ⟨ x ⟩ ^G : ⟨ x ⟩ | divides the product of r primes for every x ∈ G. In this article, we mainly study the properties of finite I(1)-groups and I(2)-groups. In particular, we get the possible upper bound of the order of their derived subgroup of such groups.

Key Words:

• Dedekind group
• Normal closure
• Soluble group

2010 AMS Subject Classification:

• 20D10

ACKNOWLEDGMENT

The authors are very grateful to the referee who read the manuscript carefully and provided a lot of valuable suggestions and useful comments.

Notes

Communicated by P. Tiep.