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(Crying)

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- Apr 17th 2010, 04:59 PMwutanggroup 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(Crying)

- Apr 17th 2010, 05:25 PMDrexel28
You have only two choices. Since we either have that or since those are, up to isomorphism, the only groups of order four. But, for if is cyclic then is abelian and so . It follows that

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