A subgroup of a finite $p$-group $G$ is powerfully embedded in $G$ if $N^p\geq[N,G]$. But if $p=2$ then $N^4\geq[N,G]$. I want to know why definition for even and odd prime is different?
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1A subgroup of - you mean $N$? – Dietrich Burde Feb 24 '25 at 11:19
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The answer is, that powerful $p$-groups already have a different definition for $p=2$. Why? This is explained in the duplicate. – Dietrich Burde Feb 24 '25 at 11:25