# Relationship between the p-radical subgroups and the parabolics in a BN-pair generality

A theorem of Quillen says that if $$G$$ is a finite Chevalley group over characteristic $$p$$, then the poset $$\mathcal{A}_p(G)$$ of nontrivial elementary abelian subgroups of $$G$$ is homotopy equivalent (I will not distinguish a poset from its order complex much) to the associated Tits building $$\Delta(G)$$.

I am interested in finding out how far this result can be generalized for a finite $$G$$ while we can still reasonably talk about an associated building $$\Delta(G)$$. I shall make this more precise.

Let me start with stuff that does not require any "B,N type" input. Quillen, in the same 1978 paper, observed that for any finite group $$G$$, if we write $$\mathcal{S}_p(G)$$ for the poset of all nontrivial $$p$$-subgroups, then the inclusion $$\mathcal{A}_p(G) \hookrightarrow \mathcal{S}_p(G)$$ is a homotopy equivalence. This can be seen by observing that the "down-set" $$\mathcal{S}_p(G)_{ is contractible if $$Q$$ is outside $$\mathcal{A}_p(G)$$, and alluding to Quillen's Theorem A.

Applying the same idea dually, Bouc proved that writing $$\mathcal{B}_p(G) := \{Q \in \mathcal{S}_p(G) : Q = O_p(N_G(Q))\}$$

(here $$O_p$$ denotes the largest normal $$p$$-subgroup), the "up-set" $$\mathcal{S}_p(G)_{>Q}$$ is contractible when $$Q \notin \mathcal{B}_p(G)$$. Therefore the inclusion $$\mathcal{B}_p(G) \hookrightarrow \mathcal{S}_p(G)$$ is also a homotopy equivalence. Thus we can work with $$\mathcal{B}_p(G)$$ instead of $$\mathcal{A}_p(G)$$. Elements of $$\mathcal{B}_p(G)$$ are often called $$p$$-radical subgroups in the finite group theory literature. The relationship of $$\mathcal{B}_p(G)$$ with the building (when there is one) is much more direct than that of $$\mathcal{A}_p(G)$$, see below.

I don't know which notions about buildings are "standard" and which are not, so I will try to set things up about them carefully without ambiguity. I will start from a general situation in which we can talk about a building and then consider restrictions to be able to compare the building with the intrinsically defined poset $$\mathcal{B}_p(G)$$.

A general situation to get a building is the following, quoted from Abramenko-Brown's book with some abuse of notation:

Definition 1. Suppose we are given a group $$G$$, a subgroup $$B$$, a Coxeter system $$(W,S)$$, and a bijection $$C \colon W \rightarrow B \backslash G/B$$ satisfying the following condition:

(B) For all $$s \in S$$ and $$w \in W$$, $$C(sw) \subseteq C(s)C(w) \subseteq C(sw) \cup C(w)$$. If we have $$l(sw) = l(w) + 1$$, then $$C(s)C(w) = C(sw)$$.

Then the bijection is said to provide a Bruhat decomposition of type $$(W,S)$$ for $$G$$.

In the presence of a Bruhat decomposition, defining $$W_J := \langle J \rangle$$ and $$P_J := \bigcup_{w \in W_J} C(w)$$ for each $$J \subseteq S$$, the $$P_J$$ turns out to actually be a subgroup and is called a standard parabolic. Moreover, their left cosets $$\Delta(G) := \{gP_J: g \in G,\, J \subsetneq S \} \, ,$$ ordered by reverse inclusion form a building (in the AB book's sense at least).

Definition 2. We say that a pair of subgroups $$B$$ and $$N$$ of a group $$G$$ is a BN-pair if $$B$$ and $$G$$ generate $$G$$, the intersection $$T := B \cap N$$ is normal in $$N$$, and the quotient $$W := N/T$$ admits a set of generators $$S$$ such that the following two conditions hold:

(BN1) For $$s \in S$$ and $$w \in W$$, we have $$sBw \subseteq BswB \cup BwB$$.

(BN2) For $$s \in S$$, we have $$sBs^{-1} \nleq B$$.

A group $$G$$ with a BN-pair indeed has a Bruhat decomposition with $$C(w) := BwB$$, and hence has an associated building which we can call $$\Delta(G)$$. The Solomon-Tits theorem holds in this generality:

Theorem (Solomon-Tits). Suppose $$(G,B,N,W,S)$$ is a group with a BN-pair whose Coxeter system is finite with longest element $$w_0$$, and write $$q_0 := |B : B \cap w_0 B w_0^{-1}|$$. Then the homotopy type of the associated building has the wedge decomposition $$\Delta(G) \simeq \bigvee_{i=1}^{q_0} \large{\mathbb{S}}^{|S|-1} \, .$$

We have not incorporated the prime $$p$$ into the definition of a BN-pair yet. That is the next step.

Definition 3. A BN-pair $$(G,B,N,W,S)$$ is called split of characteristic $$p$$ if $$G$$ is finite, and

(SBN1) $$T := B \cap N$$ equals $$B \cap w_0 B w_0^{-1}$$, and is abelian of order prime to $$p$$.

(SBN2) $$B$$ splits as a semi-direct product $$B = U \rtimes T$$ with $$U$$ a $$p$$-group.

If moreover $$U \cap w_I U w_I^{-1}$$ is normal in $$U$$ for all $$I \subseteq S$$ (here $$w_I$$ is the longest element of $$W_I := \langle I \rangle_{W}$$), then the BN-pair is called strongly split.

Writing $$U_I := U \cap w_I U w_I^{-1}$$ above, Cabanes-Enguehard show in their book that in the strongly split situation we have $$N_G(U_I) = P_I$$ and $$U_I = O_p(P_I)$$. Thus each $$U_I$$ is a $$p$$-radical subgroup of $$G$$, and we get a well-defined map $$\Phi: \Delta(G) \rightarrow \mathcal{B}_p(G)$$ defined by $$\Phi(gP_I) := gU_Ig^{-1}$$.

Now if $$gP_J \subseteq hP_I$$, then $$P_J \subseteq P_I$$ and $$gP_I = hP_I$$, so it follows from elementary results that $$gU_Ig^{-1} \subseteq hU_Jh^{-1}$$, and vice versa.

We see that, keeping in mind that the definition of $$\Delta(G)$$ has reverse inclusions, $$\Phi$$ is an order preserving map which identifies $$\Delta(G)$$ with its image in $$\mathcal{B}_p(G)$$. The obvious question is whether $$\Phi$$ is surjective. The answer is yes for the "Lie type" case:

Theorem (Borel-Tits). Let $$\mathbf{G}$$ be a simple linear algebraic group defined over an algebraically closed field of characteristic $$p \neq 0$$. Let $$\sigma$$ be an endomorphism of $$\mathbf{G}$$ onto itself such that the $$\sigma$$-fixed points $$G:= \mathbf{G}^{\sigma}$$ form a finite group. Then $$\Phi$$ is surjective.

This amounts to the last corollary of Burgoyne-Williamson's more elementary paper. The endomorphism setup is from Steinberg's work, who proved a Bruhat decomposition under these assumptions, although I couldn't find the 1968 memoirs of Steinberg online. I hope I am not misrepresenting the result, say, due to conflicting use of the term "parabolic" in these works. Also, I think Borel-Tits' original work allows to generalize to the reductive case.

Note, in particular, we now have a proof for Quillen's result I quoted in the beginning (Quillen's own argument is more direct but requires similar input): First note $$\mathcal{A}_p(G) \simeq \mathcal{B}_p(G)$$ and then Borel-Tits says $$\mathcal{B}_p(G)$$ is actually isomorphic with $$\Delta(G)$$ as a poset.

In Chapter 6 of their book, Cabanes-Enguehard tackle Alperin's weight conjecture for a finite group with a strongly split BN-pair of characteristic $$p$$, in the defining characteristic. The proof is given together with the extra assumption that $$\Phi$$ is surjective, suggesting a negative answer to my first question, in which case the second question becomes interesting:

Let $$G$$ be a finite group with a strongly split BN-pair of characteristic $$p$$.

Question 1. Is $$\Phi$$ necessarily surjective?

Question 2. Does $$\Phi$$ necessarily induce a homotopy equivalence?