Ron Solomon has been quite involved in the study of finite simple groups and their classification. He is one of the coauthors of the monumental book series on the subject, which starts with The Classification of the Finite Simple Groups. Now that we have a good classification of these groups, it is possible to look more carefully at their structure. Solomon wrote a valuable review of a paper by Salarian and Stroth where they investigate the $p$-local structure of finite simple groups. The review discusses the work that leads up to the present paper: earlier results that make the problem interesting and earlier results that make the present paper possible. By laying out the context and how the present work fits into that context, Solomon gives us a look into how group theorists are obtaining increasingly refined information about the finite simple groups, revealing them to be rich objects of study.
For finite groups, there is the notion of a group being of local characteristic $p$ for some prime, $p$. Examples are the simple groups of Lie type defined over a finite field of characteristic $p$. In this context, unlike in field theory or number theory, definitions using $p$ and the word “local” usually involve $p$-subgroups. (Remember the Sylow theorems…) The present work is connected to a larger project, led by Meierfrankenfeld, Stellmacher and Stroth (MSS), to understand the $p$-local structure of finite simple groups of local characteristic $p$ and to classify the finite simple groups of local characteristic 2. Solomon gives a quick description of that project and tells us how it fits in with other big projects, such as his own with Gorenstein and Lyons and that of Aschbacher and Smith. For those of us who are not specialists in finite groups, this review gives a nice taste of the work that is going on in the field. It is an eye-opener to see how much is known and how much remains to be done.
MR3298118
Salarian, M. Reza(IR-KHRZU2-M); Stroth, Gernot(D-MLU-IM)
Existence of strongly $p$-embedded subgroups.
Comm. Algebra 43 (2015), no. 3, 983–1024.
20D05 (20D06)
The Classification of Finite Simple Groups (CFSG) shows that most finite simple groups are groups of Lie type defined over a finite field of characteristic $p$, for some prime $p$. Each of these simple groups $G$ is of local characteristic $p$, in the sense that $C_H(O_p(H))\le O_p(H)$ for every $p$-local subgroup $H$ of $G$. (We say that a subgroup $H$ of $G$ is $p$-local if $H = N_G(P)$ for some non-identity $p$-subgroup $P$ of $G$.) Some sporadic simple groups are also of local characteristic $p$ for some prime $p$. For example, the Mathieu group, $M_{24}$, is of local characteristic 2. Indeed, many simple groups $G$, such as the alternating groups ${\rm Alt}(p)$ and the Monster (for $p\in \{ 41,47,59,71 \}$), are of local characteristic $p$ simply because they have self-centralizing Sylow $p$-subgroups of order $p$. This case may be avoided by assuming that a Sylow $p$-subgroup of $G$ lies in more than one maximal $p$-local subgroup of $G$, or by taking $p = 2$. There is an ongoing project, led by U. Meierfrankenfeld, B. Stellmacher and G. Stroth (MSS), to understand the $p$-local structure of finite simple groups of local characteristic $p$ and to classify the finite simple groups of local characteristic 2. This project is described in [U. Meierfrankenfeld, B. Stellmacher and G. Stroth, in Groups, combinatorics & geometry (Durham, 2001), 155–192, World Sci. Publ., River Edge, NJ, 2003; MR1994966 (2004i:20020)].
More recently, attention has been given to an even more ambitious project—the classification of the finite simple groups of parabolic characteristic 2. Here, $G$ is said to be of parabolic characteristic $p$ if $C_H(O_p(H))\le O_p(H)$ for every $p$-local subgroup $H$ of $G$ which contains a Sylow $p$-subgroup of $G$. Most of the sporadic simple groups are of parabolic characteristic 2, even though most are not of local characteristic 2. For example, the Monster and the Baby Monster are both of parabolic characteristic 2. We remark that the phrase `parabolic characteristic 2′, as used by MSS, is synonymous with the phrase even characteristic, as used by M. G. Aschbacher and S. D. Smith in their monographs proving the Quasithin Theorem [The classification of quasithin groups. I, Math. Surveys Monogr., 111, Amer. Math. Soc., Providence, RI, 2004; MR2097623 (2005m:20038a); The classification of quasithin groups. II, Math. Surveys Monogr., 112, Amer. Math. Soc., Providence, RI, 2004; MR2097624 (2005m:20038b)]. There is yet another weakening of the notion of local characteristic 2, called even type, which is utilized by D. Gorenstein, R. N. Lyons and R. M. Solomon in their project (GLS) to revise the proof of CFSG, as outlined in Monograph 1 of the GLS project [The classification of the finite simple groups, Math. Surveys Monogr., 40.1, Amer. Math. Soc., Providence, RI, 1994; MR1303592 (95m:20014)]. Aschbacher and Smith proved in the final chapter of their magnum opus that $\mathcal{K}$-proper quasithin simple groups of even type are all of even characteristic (parabolic characteristic 2), except for the sporadic Janko group $J_1$. In particular, 11 of the 26 sporadic simple groups are quasithin of parabolic characteristic 2. This result has recently been extended by Magaard and Stroth to an identification of all $\mathcal{K}$-proper simple groups of even type which are not of parabolic characteristic 2. I should note that a finite group $G$ is called $\mathcal{K}$-proper if every simple section of every proper subgroup of $G$ is on the list of conclusions of CFSG. Thus, a minimal counterexample to CFSG is $\mathcal{K}$-proper, and some version of this hypothesis is used in all current efforts to provide a revised proof of CFSG. In particular, the paper currently under review uses the related notion that $G$ is a $\mathcal{K}_p$-group if every simple section of every $p$-local subgroup of $G$ is a known finite simple group, i.e., appears on the list of conclusions of CFSG. The GLS project proposes to provide a revised treatment of the classification of simple groups, both of even type and of odd (= not even) type, quoting in particular the Aschbacher-Smith monographs for the treatment of quasithin groups of even type. The GLS treatment of the $\mathcal{K}$-proper simple groups of odd type will be completed in Monograph 7 of their series, of which six monographs have been published. We may hope that the even type (parabolic characteristic 2) portion of CFSG will in the near future have two largely independent treatments: AS + GLS and MSS.
We turn more specifically to the paper under review and its place in the MSS project. A fundamental role in the authors’ program is played by large subgroups. For them, a $p$-subgroup $Q$ of a group $G$ is large if
- (1) $C_G(Q)\le Q$; and
- (2) $N_G(U)\le N_G(Q)$ for all non-identity $U\le C_G(Q) = Z(Q)$.
An easy argument shows that the existence of a large subgroup implies that $G$ is of parabolic characteristic $p$. MSS have proved an important Structure Theorem [The local structure theorem for finite groups with a large $p$-subgroup, Mem. Amer. Math. Soc., Amer. Math. Soc., to appear] describing the $p$-local structure of finite groups $G$ having a large $p$-subgroup. The next step in the program is to identify the subgroup $H$ of $G$ generated by certain $p$-local overgroups of a fixed Sylow $p$-subgroup $S$ of $G$. In general $F^*(H)$ will be a simple group of Lie type in characteristic $p$ containing $N_G(X)$ for all non-identity normal subgroups $X$ of $S$. The final step is to prove that $H = G$, thereby identifying $G$. The paper under review then completes this final step in the generic case, when $G$ is of local characteristic $p$. Specifically, the authors work under the following Hypothesis 1:
- (a) There is a subgroup $H$ of the finite group $G$ such that $F^*(H) = G(r)$, $r = p^u$, is a group of Lie type and rank as a $BN$-pair at least $2$, and at least $3$ if $p$ is odd.
- (b) If $S\in Syl_p(H)$ and $1\ne X\triangleleft S$, then $N_G(X)\le H$.
They prove the following two theorems.
Theorem 1. Assume that $G$ is of local characteristic $p$ and that Hypothesis 1 holds for $G$. Then $H$ is strongly $p$-embedded in $G$, i.e., $N_G(X)\le H$ for every non-identity $p$-subgroup $X$ of $H$.
Theorem 2. Assume that $G$ is of local characteristic $p$ and that Hypothesis 1 holds for $G$. Then $G = H$ if one of the following conditions holds:
- (1) $p = 2$; or
- (2) $p$ is odd and $G$ is both a $\mathcal{K}_p$-group and a $\mathcal{K}_2$-group.
Note that groups with a proper strongly $2$-embedded subgroup were classified by H. Bender [J. Algebra 17 (1971), 527–554; MR0288172 (44 #5370)], which is why Theorem 2 follows immediately from Theorem 1 when $p = 2$. For $p$ odd, Theorem 2 follows immediately from Theorem 1 and the main theorem of [C. W. Parker and G. Stroth, Pure Appl. Math. Q. 7 (2011), no. 3, Special Issue: In honor of Jacques Tits, 797–858; MR2848592 (2012f:20071)]. Also, as noted by the authors, a version of Theorem 2 for $p = 2$ has been proved by Aschbacher in [Invent. Math. 180 (2010), no. 2, 225–299; MR2609243 (2011c:20037)] without Hypothesis 1(a), but under the hypothesis that $G$ is a $\mathcal{K}_2$-group, which the authors avoid here.
The paper under review uses Aschbacher’s local $C(G,T)$-Theorem as revised by D. M. Bundy, N. Hebbinghaus and B. Stellmacher [J. Algebra 300 (2006), no. 2, 741–789; MR2228220 (2007c:20050)], and extended to the class of groups of local characteristic $p$. In order to pass from the hypothesis that $N_G(X)\le H$ for all non-identity normal subgroups $X$ of $S$ to the conclusion that $N_G(X)\le H$ for all non-identity subgroups of $S$, the authors must use the “pushing-up” methodology. In particular, they must confront obstructions to pushing-up, dubbed blocks by Aschbacher. Here the obstructions studied are in sets $\mathcal{P}^*(X)$, arising as follows. For $X$ a $p$-subgroup of $H$, the authors set $$ \mathcal{M}(X) = \{ K : K\nleq H, O_p(K)\ne 1, X\le K \}. $$ They define a partial ordering on such subgroups $K$ based on containment of Sylow $p$-subgroups of $K\cap H$, and let $\mathcal{M}_{\max}(X)$ denote the maximal elements under this partial ordering. Then they let $\mathcal{P}(X)$ denote the minimal members of $\mathcal{M}_{\max}(X)$ under inclusion. Finally, they set $$ \mathcal{P}^*(X) = \{ K : K\in \mathcal{P}(X), O_{p’}(K)\le H, F^*(K/O_{p’}(K)) = O_{p’p}(K)/O_p(K) \}. $$ These are their “blocks”, whose structure was determined mainly by [D. M. Bundy, N. Hebbinghaus and B. Stellmacher, op. cit.]. Arguments in Sections 3, 4 and 5 narrow down the possible structures considerably, using Hypothesis 1. Next, letting $R$ denote a long (short if $F^*(H)\cong {\rm Sp}_{2n}(2^m)$) root subgroup of $F^*(H)$, the authors proceed, mostly under the hypothesis (5.1) that $C_G(t)\nleq H$ for some non-identity $t\in C_H(R)$ of order $p$. The goal is to prove that $\mathcal{P}^*(C_S(t)) = \emptyset$ for all such $t$. This is achieved in Section 6 for $p = 2$, and in Section 7 for odd $p$. However, when $p = 2$, certain possibilities for $F^*(H)$ of $BN$-rank 2 were excluded in Hypothesis 4.5. These are handled in Section 8. Finally, a fairly short argument in Section 9 now shows that $C_G(t)\le H$ for all $t\in H$ of order $p$. As $N_G(S)\le H$, an easy standard argument yields that $H$ is strongly $p$-embedded in $G$, as desired.
As noted earlier, this paper is an important component of the endgame for the MSS classification project. With a view to the possible extension of this project to cover groups of parabolic characteristic $p$, the authors avoid the hypothesis of local characteristic $p$ until Section 9 of this paper.
