Give an example of a group G and its subgroup H of index 2008 that is not normal.
What is the answer to this question, written out, and how also would i solve it if the index was different?
let $\displaystyle n \geq 3$ and $\displaystyle H=\{\sigma \in S_n: \ \sigma(1)=1 \}.$ then obviously $\displaystyle H$ is a subgroup of $\displaystyle S_n$ and $\displaystyle [G:H]=n.$ let $\displaystyle \alpha=(1 \ \ 2) \in S_n$ and choose $\displaystyle \sigma \in H$ with $\displaystyle \sigma(2) \neq 2.$ then $\displaystyle \alpha \sigma \alpha^{-1}(1)=\alpha \sigma(2) \neq 1.$
thus $\displaystyle \alpha \sigma \alpha^{-1} \notin H,$ which means $\displaystyle H$ is not normal in $\displaystyle S_n. \ \ \Box$