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Math Help - 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 a,b\in  G then 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 a'\in H and b' \in H then by (*) we have a'*b' \in H

    Also the identity belongs to H since e^2=e

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

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

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