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

Solution 1:

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.

• Weir, A. J. “Sylow $p$-subgroups of the classical groups over finite fields with characteristic prime to $p$.” Proc. Amer. Math. Soc. 6 (1955), 529–533. MR72143 DOI:10.2307/2033424
• Carter, Roger; Fong, Paul. “The Sylow 2-subgroups of the finite classical groups.” J. Algebra 1 (1964) 139–151. MR166271 DOI:10.1016/0021-8693(64)90030-4
• Wong, Warren J. “Twisted wreath products and Sylow 2−subgroups of classical simple groups.” Math. Z. 97 (1967) 406–424. MR212095 DOI:10.1007/BF01112816
• Stather, Mark. “Constructive Sylow theorems for the classical groups.” J. Algebra 316 (2007), no. 2, 536–559. MR2356843 DOI:10.1016/j.jalgebra.2007.03.013

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.

• Weir, A. J. “Sylow $p$-subgroups of the general linear group over finite fields of characteristic $p$.” Proc. Amer. Math. Soc. 6, (1955). 454–464. MR69175 DOI:10.2307/2032788
• Carter, Roger W. Simple groups of Lie type. Pure and Applied Mathematics, Vol. 28. John Wiley & Sons, London-New York-Sydney, 1972. viii+331 pp. Mr407163

Sporadic

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

Solution 2:

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

• Kondratʹev, A. S. “Normalizers of Sylow 2-subgroups in finite simple groups.” (original) Mat. Zametki 78 (2005), no. 3, 368--376; translation in Math. Notes 78 (2005), no. 3-4, 338–346 MR2227510