1. Show that if a is in G, where G is a finite group with identity e, then there exists n in positive integers such that a^n=e
2. Let G be a group and let a be one fixed element of G. Show that H (sub a)= {x in G: xa=ax}
Let and say consider . Now if for some then where . Otherwise if are all distinct then by pigeonholing it is a permutation of element and so for .
I assume you want to show is a subgroup of G. What have you done? Show what you tried to do.2. Let G be a group and let a be one fixed element of G. Show that H (sub a)= {x in G: xa=ax}
First, this is subgroup has a name. It is called the centralizer of . And it is . So basically it is the set of all that commute with all . Let then so but so this means so . This has the identity . Now it has the associative property because G which contains it has that property. Finally if then it means thus thus and so . So this proves it is a subgroup.