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I need your help in following question. I apologies for asking similar type questions. I would appreciate any help. I am sorry for the inconvenience.

I wish to classify $p$ group $G$ ($p$ prime) of order $p^4$ and $p^5$ such that whenever $H$ is a non normal subgroup of order $p$, $G$ is the semidirect product of $H$ and a subgroup $K$ with $K$ isomorphic to the quotient $G/L$ where $L$ is any normal subgroup of order $p$.

Can we classify finite $p$ group with this property.

Thanks in advance.

vipul kakkar

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What have you tried? Do you have one single example of such a group? There are several things that can be said about such $\,G\;$ : it isn't abelian, $\,|Z(G)|=p\;\;\text{or}\;\;p^2\;$ ...yet the characterization you want looks pretty specific. – DonAntonio Aug 21 '13 at 12:09
@ DonAntonio yes, if $G \cong C_p \ltimes (C_p \times C_p)$, then it satifies the hypothesis. – Vipul Kakkar Aug 21 '13 at 16:51
Your example has order $\;p^3\;$ . You asked about groups of order $\;p^4\;,\;p^5\;$ ... – DonAntonio Aug 21 '13 at 19:44
@ DonAntonio Yes, for groups of order $p^4$ and $p^5$ I could not find example. Without using classification I need simple argument whether there exists a group with such property. – Vipul Kakkar Aug 22 '13 at 4:48

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