p-groups as Sylow subgroups - MathOverflow

p-groups as Sylow subgroups

2011-02-07T13:49:14Z

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).)

Answer by Chris Godsil for p-groups as Sylow subgroups

2011-02-07T14:30:34Z

No. If the Sylow 2-subgroup of a group is cyclic then the group has a homomorphism onto its Sylow 2-subgroup. (I don't have a reference at hand, look up $p$-nilpotent groups. Or just prove that if the Sylow 2-subgroup is cyclic, there is a homomorphism on $\mathbb{Z}_2$ and use induction.) So, a few small cases aside, neither a symmetric group nor a general linear group can have a cyclic Sylow 2-subgroup.

As already noted, every $p$-group occurs as a subgroup of a Sylow p-subgroup of the symmetric group. Any group of order $n$ can be represented a permutation group of order $n$ (Cayley's theorem), hence any group of order $n$ occurs as a subgroup of $S_n$ and so so a group of order $p^k$ occurs as a subgroup of $S_{p^k}$. If you now replace permutations by permutation matrices, you can embed any group in the general linear group and so any $p$-group is a subgroup of a linear group.

Answer by Joe Shmo for p-groups as Sylow subgroups

2011-02-07T14:58:30Z

To be clear, the first answer also contradicts Pete's second assertion, as each p-subgroup of a group embeds in a Sylow p-subgroup of that group.

Answer by Jim Humphreys for p-groups as Sylow subgroups

2011-02-07T18:02:14Z

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.

Answer by F. Ladisch for p-groups as Sylow subgroups

2011-02-07T18:43:14Z

Just a remark on symmetric groups: The structure of the Sylow $p$-subgroups of $S_n$ is indeed known: The Sylow $p$-subgroup of the symmetric group on $n=p^k$ letters is
$$ \underbrace{C_p \wr C_p \wr \cdots \wr C_p}_{k}, $$ that is, an $k$-fold wreath product of cyclic groups of order $p$ . (See Huppert, Endliche Gruppen I, Satz III.15.3.) If the number of letters, $n$, is not a prime power, the Sylow subgroup is a direct product of groups of this type.
A simpler remark: There is no $n$ such that the Sylow $p$-subgroup of $S_n$ has order $p^p$, as is easy to see.
So symmetric groups are of no use here.

Answer by Geoff Robinson for p-groups as Sylow subgroups

2011-04-29T09:32:14Z

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.