Minimal Extension Covers

Abstract

Let A and B be finite groups and let S be the set of all extensions of A by B. A group G is called an extension cover of (A, B), if G contains all extensions in S as subgroups of G. A group G is called a minimal extension cover if G is an extension cover of minimal order. Let be the prime factorization of the odd number n and define . The group D _{n 1} ×…×D _{n k}  × Z ₂ is the unique minimal extension cover of (Z _n , Z ₂). This article also constructs a minimal extension cover of (Z _{2 ⁿ} , Z ₂). Some conjectures about minimal extension covers are examined as well.

Key Words:

• Extensions

2000 Mathematics Subject Classification:

• Primary 20E22
• Secondary 20D99

ACKNOWLEDGMENTS

This article was partly supported by a grant from the Faculty Research and Development committee of Towson University.

Notes

Communicated by M. Dixon.