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Math Help - subgroup

  1. #1
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    subgroup

    Let H be a group. Let M be a normal subgroup of H. Let K be any subgroup of H. Let MK={mk: m is an element of M and k is an element if K}
    a)Prove: MK is a subgroup of H.
    b)Suppose that M intersects K={I}. Let k, k' are elements of K. Prove Mk=M k' if and only if k= k'. conclude that |MK|=|M||K|
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  2. #2
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    Quote Originally Posted by apple2009 View Post
    Let H be a group. Let M be a normal subgroup of H. Let K be any subgroup of H. Let MK={mk: m is an element of M and k is an element if K}
    a)Prove: MK is a subgroup of H.
    b)Suppose that M intersects K={I}. Let k, k' are elements of K. Prove Mk=M k' if and only if k= k'. conclude that |MK|=|M||K|
    For (a), it would be helpful to use the following lemma.

    Lemma. If M and K are subgroups of a group H, MK is a subgroup of H if and only if MK = KM.

    Now you need to show that MK \subseteq KM and KM \subseteq MK in order to apply the above lemma. I'll show MK \subseteq KM and I'll leave it to you to show that the reverse inclusion.

    Let m \in M, k \in K such that mk \in MK. Since M is a normal subgroup of H, mk = k(k^{-1}mk) \in KM. Thus MK \subseteq KM.

    (b) You might need to use this.
    |MK| = \frac{(|M|)(|K|)}{|M \cap K|}
    Last edited by aliceinwonderland; November 8th 2009 at 10:13 AM.
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