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}

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- Feb 10th 2008, 03:50 PMfulltwist8Abstract: More groups
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} - Feb 10th 2008, 03:54 PMThePerfectHacker
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 .

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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}

- Feb 10th 2008, 03:58 PMfulltwist8
yes, prove that it's a subgroup. sorry, i forgot that part! i have no idea where to start on number two. is it like saying x*a = a*x?

- Feb 10th 2008, 04:08 PMThePerfectHacker
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.