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).)
[Math] p-groups as Sylow subgroups
gr.group-theory
Best Answer
The question is somewhat loosely stated, leading to various answers and comments which are at cross-purposes. Some specific families of finite groups are mentioned, but the list seems to be left open (?) Among these families, the symmetric and alternating groups have no built-in prime $p$ to favor. Moreover, Burnside's theorem (as people have noted) always allows one to embed a given $p$-group in some symmetric group; if the prime is odd, this is the same as embedding into a (usually simple) alternating group. But in such cases there would probably be only a minuscule chance that the given $p$-group can be embedded as a Sylow $p$-subgroup. I can't prove this offhand but suspect it's known to finite group theorists. As Pete suggests, some numerical estimates might be a good tool. Work by P. Hall and others has shown how rapidly the number of different groups of order $p^n$ grows for a given prime as $n$ grows; on the other hand, there is a lot of classical literature on the Sylow structure of symmetric groups.
The other families mentioned are among those of Lie type, defined relative to a specific prime $p$. Here the structure of a Sylow $p$-subgroup is severely constrained by the root system involved, whereas the order of such groups is easy enough to specify. So again it seems most unlikely outside limited cases that a given $p$-group will be on the list of Sylow subgroups even if you take all groups of Lie type into account. (This too might be in the literature.) For these families it doesn't even seem plausible to me that one can embed an arbitrary $p$-group into such a Sylow subgroup over a large enough field of characteristic $p$. But that wasn't the question asked.
Note that the study of a Lie family changes radically if one wants to say something about the Sylow $r$-subgroups for $r \neq p$. This has been an active topic in the study of modular representations, but gets extremely complicated to organize in a meaningful way.
I think the bottom line is that not much can be learned about the huge world of $p$-groups by trying to embed them in any of these familiar families of groups, even though that is of course possible in some very special cases.