# Thread: Group, subgroup H of index n that is not normal

1. ## Group, subgroup H of index n that is not normal

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?

2. Originally Posted by cjart.1
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 $n \geq 3$ and $H=\{\sigma \in S_n: \ \sigma(1)=1 \}.$ then obviously $H$ is a subgroup of $S_n$ and $[G:H]=n.$ let $\alpha=(1 \ \ 2) \in S_n$ and choose $\sigma \in H$ with $\sigma(2) \neq 2.$ then $\alpha \sigma \alpha^{-1}(1)=\alpha \sigma(2) \neq 1.$

thus $\alpha \sigma \alpha^{-1} \notin H,$ which means $H$ is not normal in $S_n. \ \ \Box$