# Math Help - Centralizer of Group

1. ## Centralizer of Group

Let G be a finite group with g in G.
Ng, the centralizer of g in G is defined to be the set = {x|xgx^-1 = g}

Show that Ng is a subgroup of G.

Now, letting Sg = {xgx^-1|x in G}

Prove that, |Sg| = |G/Ng|
and that {Sg|g in G} is a partition of G

2. Originally Posted by Coda202
Show that Ng is a subgroup of G.
Just check the definitions of being a subgroup.

Prove that, |Sg| = |G/Ng|
and that {Sg|g in G} is a partition of G
Hint: $aga^{-1} = bgb^{-1}$ iff $(b^{-1}a)g(b^{-1}a)^{-1} = g$ iff $a(Ng) = b(Ng)$.