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Math Help - iven any group G and a subset W, let < W > be the smallest subgroup of G which contai

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    iven any group G and a subset W, let < W > be the smallest subgroup of G which contai

    Given any group G and a subset W, let < W > be the smallest subgroup of G which contains W?

    A) prove that there is such a subgroup < W > in G. ( < W > is called the subgroup generated by W.)

    B) If gwg ^ -1 W for all g G, w W, prove that < W > is a normal subgroup of G.

    C) Now, let U={xyx^ -1y^-1 | x, y G}. Prove that < U > is normal in G.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by snick View Post
    Given any group G and a subset W, let < W > be the smallest subgroup of G which contains W?

    A) prove that there is such a subgroup < W > in G. ( < W > is called the subgroup generated by W.)

    B) If gwg ^ -1 W for all g G, w W, prove that < W > is a normal subgroup of G.

    C) Now, let U={xyx^ -1y^-1 | x, y G}. Prove that < U > is normal in G.
    A) Let \mathcal{M}=\left\{H:W\subseteq H\leqslant G\right\}. Define K=\bigcap_{M\in\mathcal{M}}M

    B) What definition do you use?

    C) What have you tried?
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