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Math Help - Group, subgroup H of index n that is not normal

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    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?
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    Quote Originally Posted by cjart.1 View Post
    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
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