Answer to Question #140394 in Abstract Algebra for J

Question

state what properties you use/defintions/etc...

Let (G,✳) be a group, and let a∈G. Let C(a) = {g∈G: a✳g = g✳a}.

In this problem we will prove that (C(a),✳) is a subgroup of (G,✳).

C(a)⊆G by the definition of C(a).

Answer 1

Solution

$C(a)$ is not empty since $\exist e \in G$ such that $a *e=e*a$ so $e\in C(a)$ (where e is the identity element of G)

Since $C(a)\subseteq G$ then $C(a)$ is associative under the operation $*$ .

Let $x,y\in C(a)$, then $a*x=x*a$ and $a*y=y*a$

$a*x*y=a*(x*y) =(x*y)*a$ (by definition) so $(x*y)\in C(a)$

Since $e \in C(a)$ we have $a*e=a*(x*x^{-1}) = (a*x)*x^{-1}$ and $e*a=(x^{-1}*x)*a= x^{-1}*(x*a)$ which implies that $x^{-1}\in C(a)$

Therefore $(C(a),*)$ is a subgroup of the group $(G,*)$ .