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Let $G$ be a finite group such that $G/Z(G)\cong K_4$ (Klien's 4 group).

$G/G'$ and $Z(G)$ both have exponent $4$, then $G$ can be written as direct product of $A$ times $H$ where $A$ is an abelian group and $H$ cannot be written as such a direct product as G is written means H is indecomposable. In other words we can say that H behaves similar to G means same properties as that of G but how to prove that G can be written in such a way?

PAMG
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  • Question is not clear: every group $G$ can be factored into an abelian part $A$ and a non-abelian part $H$, in which $H$ is purely non-abelian (i.e. it has no abelian factor). Make question precise. – p Groups Jan 12 '16 at 04:31
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    Can't every finite group $G$ be written as a direct product $G=A\times H$ where $A$ is abelian and $H$ can't be written as such a product in a non-trivial way? (Doesn't that follow from the uniqueness of the decomposition?)

    http://math.stackexchange.com/questions/908033/uniqueness-of-the-direct-product-decomposition-of-finite-groups

     – verret Jan 12 '16 at 05:52
  • What do you mean by $H$ has the same properties as $G$? Can you say exactly what properties you want $H$ to have? Also, I think the second paragraph should start "If $G/G'$ and $Z(G)$ both have ...". – Derek Holt Jan 12 '16 at 15:24
  • means H/Z(H) is isomprophic to K4 and again Z(H) has exponent 4 also H/H' – PAMG Jan 13 '16 at 04:07

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