Relative Covering Groups

Abstract

Let ν and ω be two varieties of groups defined by the sets of laws V and W, respectively. We introduce the concept of a ω-ν-covering group of a given group and show that every two ω-ν-covering groups of a given group in ω are ν-isologic. Also, in the case ν ⊆ ω, we show the existence of such groups for a finite ν-perfect group in ω, and also that every automorphism of a finite ν-perfect group G in ω may be lifted to an automorphism of a ω-ν-covering group of G. Finally we show that if G is in ω ∩ ν, then all ω-ν-covering groups of G are Hopfian.

Key Words:

• ω-ν-Covering group
• ν-Perfect group
• Baer-invariant
• Isologism
• Hopf property

2000 Mathematics Subject Classification:

• 20E10
• 20E34
• 20E36

Acknowledgment

We would like to express our sincere thanks and gratitude to Professors Robert G. Burns, Akbar Rhemtulla and Mazi Shirvani who read the manuscript and made valuable suggestions.

This work was done while the first author visiting York University, and he would like to thank the hospitality of Mathematics and Statistics Department.