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About LinkBacksIf 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. SinceOriginally Posted by wutang
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 helpwe 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 orderwhere
is a prime is abelian and the rest follows from the fundamental theorem of finitely generated abelian groups.
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