Centralizer Math Help: Proving <a> is a Subset of C(a) for Abelian Groups

[kathrynag]

Homework Statement

Show <a>\subseteqC(a) where C(a) is such that xa=ax.

Homework Equations

The Attempt at a Solution

Knowing xa=ax tells me we have an abelian group.

 

[kathrynag]

Also a is in G

 

[kathrynag]

<a> is a cyclic subgroup

 

[kathrynag]

we have x=a^n . Need to show x=a^n is a subset of xa=ax. Not sure how to do that

 

[kathrynag]

So x=aaaaaaaaaa
a^-1x=a^n-1
I have no clue where my thought process has gone wrong, but it's really frustrating me

 

[jbunniii]

You have to show that for every n, a^n commutes with a. There's almost nothing to prove.

 

[HallsofIvy]

Homework Statement

Show <a>\subseteqC(a) where C(a) is such that xa=ax.

Homework Equations

The Attempt at a Solution

Knowing xa=ax tells me we have an abelian group.

No, it doesn't. It tells you that C(a) is the set of all members of G that commute with a. If G is commutative, then C(a)= G. Otherwise C(a) is a subset of G.

Now, exactly what is the definition of <a>. Yes, I know it is the cyclic subgroup generated by a, but exactly what is that? Note that it is common to use either "multiplicative" notation, ab, or "additive" notation, a+ b, for the group operation. How would <a> be written in those two notations?

 

[Hurkyl]

No, it doesn't. It tells you that C(a) is the set of all members of G that commute with a. If G is commutative, then C(a)= G. Otherwise C(a) is a subset of G.

If you meant "proper subset", you have to further assume that a is not in the center of G.