Cyclic, Dicyclic and Metacyclic Groups

Abstract

After briefly defining such fundamental concepts as generators, factor groups and direct products, we show how an automorphism of a given group enables us to adjoin a new element so as to obtain a larger group; e.g., the cyclic and non-cyclic groups of order 4 yield the quaternion group and the tetrahedral group, respectively. Observing that the standard treatises use the term metacyclic group in two distinct senses, we exhibit both kinds among the groups of order less than 32, whose simplest known abstract definitions are collected in Table 1.