List questions from p-groups

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Let $G$ be non-abelian, $|G|=p^3$, where $p$ is an odd prime. Show $\exists N\unlhd G$ with $Z(G)<N<G$ and $N\cong\Bbb Z_p \times\Bbb Z_p$.

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A finite group of order $p^n$ cannot have a center of order $p^{n-1}$

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Center of $p$-group of maximal class

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Nilpotency class of any epimorphic image of a p-group of maximal class

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If $f$ is a homomorphism, does it map $p$-groups to $p$-groups?

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Sylow subgroups of soluble groups

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Order and element set of the group with presentation $\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$

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Subgroups of abelian $p$-groups

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$\exists a\in G-H$ such that $aHa^{-1}=H$

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Classify $p$-groups in which all groups of the same order are isomorphic

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$|G|=p^3$ non abelian $\Longrightarrow\gamma_2(G)\lneq Z(G)$

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Proving group is $p$-group by contradiction

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Cauchy's Theorem $\#$ of elements order $p$

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A list of conjugacy classes in elementary abelian $p$-group

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List Question

Subgroups of infinite $p$-group ($p$ is prime)

Let $G$ be non-abelian, $|G|=p^3$, where $p$ is an odd prime. Show $\exists N\unlhd G$ with $Z(G)<N<G$ and $N\cong\Bbb Z_p \times\Bbb Z_p$.

A finite group of order $p^n$ cannot have a center of order $p^{n-1}$

Center of $p$-group of maximal class

Nilpotency class of any epimorphic image of a p-group of maximal class

If $f$ is a homomorphism, does it map $p$-groups to $p$-groups?

Sylow subgroups of soluble groups

Order and element set of the group with presentation $\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$

Subgroups of abelian $p$-groups

$\exists a\in G-H$ such that $aHa^{-1}=H$

Classify $p$-groups in which all groups of the same order are isomorphic

$|G|=p^3$ non abelian $\Longrightarrow\gamma_2(G)\lneq Z(G)$

Proving group is $p$-group by contradiction

Cauchy's Theorem $\#$ of elements order $p$

A list of conjugacy classes in elementary abelian $p$-group

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