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 $2$).
- Weisner, Louis;
“On the Sylow Subgroups of the Symmetric and Alternating Groups.”
Amer. J. Math. 47 (1925), no. 2, 121–124.
MR1506549
DOI:10.2307/2370639
- Kaloujnine, Léo
“La structure des p-groupes de Sylow des groupes symétriques finis.”
Ann. Sci. École Norm. Sup. (3) 65, (1948). 239–276.
Also see: C. R. Acad. Sci. Paris
221 (1945), 222–224; ibid.
222 (1946), 1424–1425; ibid.
223 (1946), 703–705; ibid.
224 (1947), 253–255.
- Weir, A. J.
“The Sylow subgroups of the symmetric groups.”
Proc. Amer. Math. Soc. 6 (1955), 534–541.
MR72142
DOI:10.2307/2033425
- Lewis, Robert Edward.
“On the Sylow two-subgroups of the alternating groups.”
Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. 1968. 48 pp.
MR2617989
- Dmitruk, Ju. V.; Suščanskʹkiĭ, V. Ī.
“Construction of Sylow 2-subgroups of alternating groups and normalizers of Sylow subgroups in symmetric and alternating groups.”
Ukrain. Mat. Zh. 33 (1981), no. 3, 304–312.
MR621637
- Harada, Koichiro; Lang, Mong Lung.
“Indecomposable Sylow 2-subgroups of simple groups.”
Acta Appl. Math. 85 (2005), no. 1-3, 161–194.
MR2128910
DOI10.1007/s10440-004-5618-0
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.
- Aschbacher, Michael.
“A characterization of Chevalley groups over fields of odd order.”
Ann. of Math. (2) 106 (1977), no. 2, 353–398.
MR498828
- Aschbacher, Michael.
“A characterization of Chevalley groups over fields of odd order. II.”
Ann. of Math. (2) 106 (1977), no. 3, 399–468.
MR498829
- Aschbacher, Michael.
“On finite groups of Lie type and odd characteristic.”
J. Algebra 66 (1980), no. 2, 400–424.
MR593602
DOI:10.1016/0021-8693(80)90095-2
- Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald.
The classification of the finite simple groups. Number 3. Part I. Chapter A.
Almost simple K-groups. Mathematical Surveys and Monographs, 40.3. American Mathematical Society, Providence, RI, 1998. xvi+419 pp. ISBN: 0-8218-0391-3
MR1490581
- Harada, Koichiro; Lang, Mong Lung.
“Indecomposable Sylow 2-subgroups of simple groups.”
Acta Appl. Math. 85 (2005), no. 1-3, 161–194.
MR2128910
DOI10.1007/s10440-004-5618-0
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.
Best Answer
Here are my notes, since I like fairly explicit answers.
In case anyone wants to see the other examples more briefly:
Elementary abelian p-groups of order pn are found TI in PSL(2,pn). Such subgroups are uniquely identified by which subspace they stabilize.
Quaternion 2-groups are found TI in the their semi-direct products with their (unique) faithful irreducible module over a prime field of odd order. The key point is that the unique element of order 2 acts fixed-point-freely on the module, and so is not contained in the intersection of any two distinct Sylow 2-subgroups.
A weaker question has a positive answer:
Yes. For any p-group P, take G to be the semi-direct product of P with a large enough faithful module V over a finite field of characteristic not p. Then consider the union of CV(x) as x varies over P. Since V is faithful, each centralizer is a proper subspace, and if V has large enough dimension, it cannot be written as the union of |P| proper subspaces. If v is some element of V outside that union of centralizers, then P ∩ Pv = 1.
For p = 2, the classification is due to Suzuki (1964) who discovered his infinite family of finite simple groups in this same line of investigation. Not only did he classify the possibly P, but also the possible G. Let N be the largest odd-order normal subgroup of G.
In other words, I missed two infinite families (and one fusion system). The structure of N is restricted, but not I think not classified. In some sense, one should read this list as "there exists an N such that...".
The classification for odd p appears to be a post-CFSG result, and is closely related to strongly embedded subgroups (similar to the concept Arturo mentions below: malnormal). The strongly embedded list is on page 383 - 384 of number 3 of the GLS writeup of the CFSG. The TI list is in Blau-Michler (1990), but is based on older lists I have not yet tracked down.
Other than cyclic and elementary abelian, there are a few more infinitely families, and a few sporadic exceptions.
Bibliography:
Suzuki, Michio. "Finite groups of even order in which Sylow 2-groups are independent." Ann. of Math. (2) 80 (1964) 58–77. MR162841 DOI:10.2307/1970491
Blau, H. I.; Michler, G. O. "Modular representation theory of finite groups with T.I. Sylow p-subgroups." Trans. Amer. Math. Soc. 319 (1990), no. 2, 417–468. MR957081 DOI:10.2307/2001249