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Thread: prove or disprove

  1. #1
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    prove or disprove

    Let G be an Abelian group. Prove or disprove that
    H={g^2|g is a member of G}
    is a subgroup of G.
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  2. #2
    Super Member PaulRS's Avatar
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    Indeed it's true, take $\displaystyle a,b\in G$ then $\displaystyle a^2*b^2=a*a*b*b=a*b*a*b=(a*b)*(a*b)=(a*b)^2$ (*)

    Where we've used the fact that G is abelian and that the associativity holds.

    So if $\displaystyle a'\in H$ and $\displaystyle b' \in H$ then by (*) we have $\displaystyle a'*b' \in H$

    Also the identity belongs to H since $\displaystyle e^2=e$

    And each element has an inverse, say we have $\displaystyle a' \in H$ then $\displaystyle a'=a^2$ for some $\displaystyle a \in G$

    Now $\displaystyle a$ has an inverse, namely $\displaystyle a^{-1} \in G$

    So note that $\displaystyle a^2*a^{-2}=a*a*a^{-1}*a^{-1}=e$ thus $\displaystyle a^{-2}=(a^{-1})^2\in H$ is the inverse of a'
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