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    group theroy question

    If G is a group and [G:Z(G)]=4, prove that G/Z(G) isomorphic to Z2+Z2. Should I look at the cosets formed by G/Z(G) or what? I hve no clue, need help
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by wutang View Post
    If G is a group and [G:Z(G)]=4, prove that G/Z(G) isomorphic to Z2+Z2. Should I look at the cosets formed by G/Z(G) or what? I hve no clue, need help
    You have only two choices. Since [G:\mathcal{Z}(G)]=4 we either have that G/\mathcal{Z}(G)\cong \mathbb{Z}_4 or G/\mathcal{Z}(G)\cong \mathbb{Z}_2\oplus\mathbb{Z}_2 since those are, up to isomorphism, the only groups of order four. But, G/\mathcal{Z}(G)\ncong\mathbb{Z}_4 for if G/\mathcal{Z}(G) is cyclic then G is abelian and so \left|G/\mathcal{Z}(G)\right|=1. It follows that G/\mathcal{Z}(G)\cong \mathbb{Z}_2\oplus\mathbb{Z}_2\cong\mathcal{K}_4

    P.S. If you aren't sure why those are the only two groups you need merely note that any group of order p^2 where p is a prime is abelian and the rest follows from the fundamental theorem of finitely generated abelian groups.
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