Problem: Let G be a group, prove that the centralizers of G is a subgroup of G.
Proof: By definitions, the centralizers of G,
Now, the identity element, e, has the property of , thus e is in C(a). So C(a) is not an empty set.
Assume C(a) contains more than just {e}, since if that is the case C(a) would be a subgroup.
Let x,y be in C(a), then:
Then
Thus xy is in C(a).
Now,
So x^{-1} is in C(a), thus C(a) is a subgroup of G.
Q.E.D.
Now, I am not sure if I have proven the last part correctly, that is, the inverse of x is in C(a), would anyone please have a look?
Oh, and the test is tomorrow morning, normally I would never ask for a free answer, as I would like to work it out myself if at all possible. But would anyone please give me the correct answer if I'm wrong as this problem might show up in the test? If you aren't comfortable with it, I fully understand, I really do appreciate the help I'm getting from here, thanks!
K