Index of Group and Subgroup

Question

Let's say we have a group $G$ and $a$ is an element of $G$ and we know that the is the set of $b$ such that $a^{-1}ba=a$ and we know that such $S$ is a subgroup of $G$ which is called centralizer of $a$. The index $(G:S)= \lvert \lbrace bab ; b \in G \rbrace \rvert$. Thanks for the help.

Answer 1

Lagrange's Theorem https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory) will help you solve your problem.

$$[G:S] = |G|/|S|$$

Here $S$ is called the center of the group.

Answer 2

$f:\{bab^{-1}:b\in G\}\to G/S$ with $f(bab^{-1})=bS$

Show that $f$ is well defined $1-1$ and onto.

Then you will have that $|\{bab^{-1}:b\in G\}|=|G/S|=[G:S]$