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What do Sylow 2-subgroups of finite simple groups look like?

It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision many answers. I tried to give an answer of "broad strokes with references." It would be reasonable to answer just for a single infinite family of finite simple groups too. Here are some specific aspects that could be addressed.

  • Which finite simple groups have the same Sylow 2-subgroup?
  • Are there ways in which Sylow 2-subgroups of finite simple groups are vastly different from a "random" or "typical" 2-group? (For instance, are the nilpotency class and derived length related in some special way compared to the general 2-group.)
  • What do the normalizers of the Sylow 2-subgroups look like (in cases where we understand the Sylow itself well)?
  • What are its characteristic, normal in the normalizer, or normal subgroups?
  • What are the possible fusion systems?
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    $\begingroup$ Jack, great question and surprisingly great answer ... by yourself? $\endgroup$ – Nicky Hekster May 9 '14 at 10:41
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    $\begingroup$ For subquestion 2: Goldschmidt's 1968 thesis shows that that the Sylow 2-subgroup of a finite simple group has exponent bounded by a function of its nilpotence class (which is unlike some annoying 2-groups like cyclic groups, in which the exponent can be quite large). I'll guess I'll wait to post an "internal structure" answer until I have more. $\endgroup$ – Jack Schmidt May 21 '14 at 20:00
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Nonabelian finite simple groups come in a few types:

  • Alternating groups
  • Classical groups in odd characteristic
  • Exceptional groups in odd characteristic
  • Groups in even characteristic (classical or exceptional)
  • Sporadic groups

In cases 1,2,3 the Sylow 2-subgroups are (slightly deformed versions of) direct products of wreath products $P_2 \wr C_2 \wr C_2 \wr \ldots \wr C_2$ where $P_2$ is the Sylow $2$-subgroup of a tiny group from the family. In case 4, the groups are best understood using linear algebra. In case 5, it would be nice to know which sporadics “borrow” a Sylow 2-subgroup and which have their own unique Sylow 2-subgroup.

Alternating

The Sylow 2-subgroups of the symmetric groups are direct products of wreath products of Sylow 2-subgroups of $S_2$ -- this was known in the 19th century. The Sylow 2-subgroups of the alternating groups are index 2 subgroups.

For $n=4m+2$ and $n=4m+3$, the copies of $S_{4m}$ inside $A_n$ have odd index $2m+1$ or $(4m+3)(2m+1)$, so the Sylow 2-subgroup of $S_{4m}$ is isomorphic to the Sylow 2-subgroups of $A_{4m+2}$ and $A_{4m+3}$.

Weisner (1925) computes the order of the normalizers of the Sylow $p$-subgroups of symmetric and alternating groups (so counts them). The main result for us is that Sylow 2-subgroups are self-normalizing in simple alternating groups (except $A_5$ with normalizer $A_4$).

Weir (1955) computes the characteristic subgroups of the Sylow $p$-subgroup of the symmetric groups, but only for odd $p$. Lewis (1968) modifies this to handle $p=2$ for both symmetric and alternating groups. Dmitruk–Suščanskʹkiĭ (1981) take the approach of Kaloujnine (1945-1948), again handling $p=2$ and alternating groups.

Harada–Lang (2005) observes that the Sylow 2-subgroups of $A_{4m}$ and $A_{4m+1}$ are directly indecomposable (while those of $A_{4m+2}$ and $A_{4m+3}$ are directly indecomposable iff $m$ is a power of $2$).

Classical groups in odd characteristic

There is a huge difference in Sylow $p$-subgroup structure depending on whether $p$ is the characteristic of the field. In this section we assume $p$ is not the characteristic of the field.

In case $p$ is not the characteristic, then Weir (1955) showed that symmetric groups and classical groups are very similar, but again $p=2$ was left out until Carter-Fong (1964), and then more uniformly in Wong (1967). Algorithms to handle all Sylow $p$-subgroups of classical groups are described in Stather (2008).

The gist is that in GL, GO, GU, and Sp, the Sylow $p$-subgroups are direct products of wreath products of cyclic groups of order $p$ with the Sylow $p$-subgroup of the two-dimensional groups. For SL, SO or $\Omega$, SU the answers are more complicated, but only because an easy to understand part has been chopped off the top.

Exceptional groups in odd characteristic

Sylow 2-subgroups for finite groups of Lie type are similar to the classical case: there is a 2-dimensional group $P_2$ and a “top” group $X$ (which need not be $C_2 \wr C_2 \wr \ldots \wr C_2$, but that is probably the correct picture to have) such that the $X$-conjugates of $P_2$ are commute with each other, so that $X \ltimes P_2^n$ is a Sylow 2-subgroup. The $P_2$ are the Sylow 2-subgroups of the so-called “fundamental subgroups” of Aschbacher (1977), where we view groups of Lie type as built up from rank 1 groups, in this case commuting rank 1 subgroups isomorphic to SL2. These are used in Aschbacher (1980) to describe groups in which a Sylow 2-subgroup is contained in a unique maximal subgroup, and Harada–Lang (2005) describes which Sylow 2-subgroups are indecomposable. GLS I.A.4.10 covers Aschbacher's ideas as well.

Groups in characteristic 2

Here the Sylow 2-subgroups are basically groups of upper triangular matrices and are often best understood in terms of linear algebra. Weir (1955) describes the characteristic subgroups and those normalized by important subgroups of GL. These general ideas work in all the groups of Lie type. The main description I know is Chevalley's commutator formula, as explained in Carter (1972).

XXX: Decent reference for the classical, and then the exceptional. Maybe specifically handle Suzuki and Ree.

Sporadic

I think each one is a special snowflake. XXX: Lookup coincidences in Sylow structure.

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Normalizers and top fusion

Kondratʹev (2005) contains a very clear description of the normalizers of Sylow 2-subgroups $S$ of finite simple groups. Typically there is not much going on: $N_G(S) = S$ unless:

  • $G$ is a group of Lie type in even characteristic (then $N_G(S)$ is a Borel subgroup and $N_G(S)/S$ is a maximally-split maximal torus, which is usually not the identity unless $q=2$)
  • $S$ is abelian, so described in Walter's theorem (Janko-Ree or certain PSL2)
  • $N_G(S)=S C_G(S)$ as in PSL, PSU, E6 and 2E6
  • $N_G(S)/S$ is an elementary abelian 3-group ($J_2,J_3, HN,PSp_{2m}$ for most values of $m$)

So other than groups of even characteristic, the automorphisms of $S$ induced from $G$ are very limited, usually inner, but with an order 7 in the Janko–Ree case and order 3 in the Janko–Ree, three more sporadics, and most of the symplectic groups.

Bibliography

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