Let $p$ be a prime number, $q$ a power of $p$ and $P$ be a finite $p$-group. $P$ is isomorphic to a subroup of p-Sylow subgroup of

  • the symmetric group $S_{\mid P\mid}$ (Theorem of Cayley)

  • the general linear group $GL(\mid P\mid, GF(q))$ (a theorem of ?)

  • the unit group $U(GF(q)P)$ of the modular group algebra $GF(q)P$ (based on a theorem of Wallace).

My question is whether there are some more non-trivial series of $p$-Sylow subgroups which let $P$ be embedded as a subgroup, too. All of them are grandfathers of all $p$-groups.

My second question is if there are some connections between these grandfather series.


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