Splitting in finite metacyclic groups.

摘要

It is well known that all finite metacyclic groups have a presentation of the form G=&angl0;x,a&vbm0; xm=1,xs at=1,ax=ar&angr0; The primary question that occupies this dissertation is determining under what conditions a group with such a presentation splits over the given normal subgroup &angl0;a&angr0; . Necessary and sufficient conditions are given for splitting, and techniques for finding complements are given in the cases where G splits over &angl0;a&angr0; . Several representative examples are examined in detail, and the splitting theorem is applied to give alternate proofs of theorems of Dedekind and Blackburn.