The Unipotent Condition of Linear Groups Generated by Matrices with Primitive Elements Jorden Blocks of Orders no More Than Eight

Abstract

In this paper, the combinatorial properties of free group generators are studied with help of calculation tool, matrix logarithm, and Jacobson radical. The unipotency of matrix groups can be better delineated by these new combinatorial properties, and a new necessary and sufficient unipotent condition of matrix group whose primitive elements Jordan blocks have orders no more than eight, as well as of free groups, will be given. Our result improves the relative theories.