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Math Help - Proving subgroup statements

  1. #1
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    Proving subgroup statements

    Suppose that H and K are subgroups of a group G. Prove the following statements under additional conditions:

    (a). First, give an example that HK may not be a subgroup of G.
    (b). Prove that if H is a normal subgroup of G, then HK is a subgroup of G.
    (c). Prove that if both H and K are normal subgroups of G, then HK is a normal subgroup of G.
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  2. #2
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    Re: Proving subgroup statements

    Quote Originally Posted by kellsbells92 View Post
    Suppose that H and K are subgroups of a group G. Prove the following statements under additional conditions:
    (a). First, give an example that HK may not be a subgroup of G.
    (b). Prove that if H is a normal subgroup of G, then HK is a subgroup of G.
    (c). Prove that if both H and K are normal subgroups of G, then HK is a normal subgroup of G.
    We are here to help you not to do the work for you.
    Therefore, we need to see what effort you have made.
    I strongly encourage to do part (a) on your own. It can teach you so much.

    If J\subseteq G~\&~\{a,b\}\subset J then if you can show that ab^{-1}\in J that is sufficient to show that J is a subgroup.
    Now once you have done (b) then (c) falls out.
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