The center of Sylow $p$-subgroups of a finite simple group of Lie type

Question

Would some one please to introduce me an easy reference!! which contains the size of $Z(P)$(center of $P$), where $P$ is a Sylow $p$-subgroup of a finite group of Lie type over a finite field of characteristic $p$?

Answer 1

I'll list some references, and give some (partial) calculations.

References:

If you have just started learning finite group theory, then I recommend Alperin–Bell's Groups and Representations. It teaches the basic ideas and uses GL as the standard example group (instead of the incredibly special dihedral groups like many early books). GL is extremely similar to nearly all finite simple groups (not sporadics, not most alternating, but otherwise, surprisingly close).

If you are still learning finite group theory, then I recommend Suzuki's Group Theory volumes 1 and 2. They teach both the finite group theory and the specific cases of some finite simple groups at the same time.

I learned from other books (Gorenstein's Finite Groups, Kurzweil–Stellmacher's Theory of Finite Groups, and the much longer Huppert–Blackburn Endliche Gruppen and Finite Groups volumes 2 and 3; I was very happy when Isaacs's Finite Group Theory appeared), but I don't think any of them integrate examples as well as Suzuki does.

If you only want to learn about the example groups (still check out Suzuki), then Wilson's Finite Simple Groups goes through the finite simple groups case by case (leaving out a bunch of cases, but you probably won't mind, since he covers so many so well). However, the unifying theme is: each case can be handled in an easy, special way.

If you want to learn about the vast majority of finite simple groups in a unified way, then Carter's Simple Groups of Lie Type is great. Some people prefer Steinberg's Lectures on Chevalley Groups.

If you want an even smoother unification (for instance to deal with harder questions that will eventually need a case-by-case analysis, but you'd like to get much farther first), then I recommend Malle–Testerman's Linear Algebraic Groups and Finite Groups of Lie Type and Carter's Finite Groups of Lie Type (which is much more advanced than his Simple book). These use a wide variety of mathematics, so are not very easy for a beginner.

Calculations:

The center of a maximal unipotent subgroup of an untwisted group of Lie type over the field $K$ is isomorphic to the additive group of $K$ unless:

• The type is $F_4$, $B_n$, or $C_n$ and $\operatorname{char}(K)=2$
• The type is $G_2$ and $\operatorname{char}(K)=3$

in which case the center is isomorphic to the additive group of $K \times K$. [ This is just a case-by-case analysis of the Chevalley commutator formula; the exceptions occur only there is a subsystem of type $B_2$ (and then $p=2$ can give trouble) or $G_2$ (and then $p=3$ can give trouble). The trouble is never very big. ]

For finite groups $G$ of Lie type:

• $A_n(q)$,
• $B_n(q)$ [$n\geq 2$, $q$ odd],
• $C_n(q)$ [$n\geq 3$, $q$ odd],
• $D_n(q)$ [$n\geq 4$],
• $G_2(q)$ [$0 \not\equiv q \mod 3$],
• $F_4(q)$ [$q$ odd],
• $E_6(q)$,
• $E_7(q)$,
• $E_8(q)$

the center of a Sylow $p$-subgroup $Q$ (for $p$ dividing $q$) is an elementary abelian group of order $q$ with no non-identity proper subgroups normalized by $N_G(Q)$.

For finite groups of Lie type:

• $B_n(q)=C_n(q)$ [ $n\geq 2$, $q$ even ]
• $G_2(q)$ [ $q=3^f$ ]
• $F_4(q)$ [ $q$ even ]

the center of a Sylow $p$-subgroup $Q$ (for $p$ dividing $q$) is an elementary abelian group of order $q^2$ which is a direct sum of two elementary abelian subgroups of order $q$, both normalized by $N_G(Q)$, but those are the only two non-identity, proper subgroups of $Z(Q)$ normalized by $N_G(Q)$.

I have not addressed twisted finite groups of Lie type:

• ${}^2A_n(q)$ [$n \geq 2$]
• ${}^2D_n(q)$ [$n \geq 4$]
• ${}^3D_4(q)$
• ${}^2E_6(q)$

(but I think $Z(Q)$ is elementary abelian of order $q$)

nor very twisted finite groups of Lie type:

• ${}^2B_2(2^{2f+1})$ [ $f \geq 1$ ]
• ${}^2G_2(3^{2f+1})$ [ $f \geq 1$ ]
• ${}^2F_4(2^{2f+1})$ [ $f \geq 1$ ]

(but I think $Z(Q)$ is elementary abelian of order $q=p^{2f+1}$)

or the Tits group:

• ${}^2F_4(2)'$, but $Z(Q)$ is cyclic of order $q=2$ from a direct computer calculation.

The question (was not asked for and) does not make sense for the other finite simple groups (alternating and sporadic).