# Math Help - Centralizer proof

1. ## Centralizer proof

Show <a> is a subset of C(a) where C(a) is such that xa=ax.

Knowing xa=ax tells me we have an abelian group.
<a> is a cyclic subgroup
we have x=a^n . Need to show x=a^n is a subset of xa=ax. Not sure how to do that

2. I don't know, but this is really frustrating me.
so x=a^n
or x=a*a*a*a*a*a.........
Since <a> is a subgroup, inverses exist
Thus x=a^-1aaaaaaa.....
x=a^n-1
We eventually get x=1
I'm not sure where I'm going wrong in my thought process

3. Originally Posted by kathrynmath
I don't know, but this is really frustrating me.
so x=a^n
or x=a*a*a*a*a*a.........
Since <a> is a subgroup, inverses exist
Thus x=a^-1aaaaaaa.....
x=a^n-1
We eventually get x=1
I'm not sure where I'm going wrong in my thought process

$\subset C_G(a)\Longrightarrow a^k\cdot a=a\cdot a^k\,,\,\forall k$ ...but this is obvious, ain't it?

Tonio