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A Tits System is defined to be a quadruple, $(G,B,N,S)$ consisting of $G$ a group, $B$ and $N$ subgroups of $G$, and $S$ a subset of $W:=N/(N\cap B)$ satisfying the following properties:

a) $B\cup N$ generates $G$ and $B\cap N$ is normal in $N$.

b) The group $W$ is generated by $S$ which consists of elements of order $2$.

c) We have $sBw\subset BwB\cup BswB$ for $s\in S$, $w\in W$.

Remark: $s$ and $w$ are really just classes modulo $N\cap B$ however as sets $sBw, BwB, BswB$ do not depend on the representatives chosen; so the above axiom does make sense.

d) For every $s\in S$ we must have that $sBs$ is not contained in $B$.

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My question is about axiom $c$. I'm confused about the remark. In the set containment claim, are we taking $s$ to be an element of $S$, or a representative of an element in $S$?

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Notation such as $sBw$ means the set of all products $sbw$ where $b \in B$. The product is happening in the group $G$, so strictly speaking, for this to make sense one needs $s \in G$. The remark is just saying that if $s'$ is congruent to $s$ modulo $B \cap N$ then $s' B w=s B w$, and thus $sBw$ depends only on the class of $s$ modulo $N \cap B$. Similarly for $w$.

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    The problem I have with Jack's comment is that if we are just considering the product of subsets then the remark is pointless. There is no need to bring representatives into the discussion. Also, if we are only insisting upon inclusion when talking about sets, then it is a weaker statement so it should matter as far as I understand.2012-10-11