TITLE: p-groups as Sylow subgroups
QUESTION [8 upvotes]: There are some nice families of groups as $S_n, A_n$, $GL(n,q)$, $SL(n,q)$, and they are useful; we know their elements, and we can  get small groups as subgroups of these groups. Is it possible to get every $p$ group as a Sylow-p subgroup of some group in such families of groups?  ( For example, the non-abelian groups of order 8 are Sylow-2 subgroups of SL(2,3) and S4; there are five non-abelian groups of order 16, having no element of order 8, and one of them, namely $D_8 \times C_2$, is Sylow-2 subgroup of $S_6$. Also $SD_{16}$ is Sylow-2 subgroup of GL(2,3).)

REPLY [2 votes]: The answer to the original question is a definite no. There are many $p$-groups which do not occur
as Sylow $p$-subgroups of classical groups, symmetric groups, or close relatives. As noted in 
various comments, embedding a finite $p$-group in a symmetric or classical group is an easy matter.
But it is very difficult for a finite $p$-group $P$ to be a Sylow $p$-subgroup of a group $G$ with no factor group of order $p$. The most general results in this direction are probably by George
Glauberman. For example, when $p \geq 5$ and $P$ is a finite $p$-group whose outer automorphism
group is a $p$-group, then any finite group $G$ with $P$ as a Sylow $p$-subgroup has a factor
group of order $p$. There are meaningful senses in which ``most" finite $p$-groups have outer
automorphism group a $p$-group.