# Centralizer proof

• Oct 21st 2010, 08:05 PM
kathrynmath
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
• Oct 21st 2010, 09:06 PM
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
• Oct 21st 2010, 09:46 PM
tonio
Quote:

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

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

Tonio