I was reading this writeup on the Mathieu groups, and got stuck on a statement in page 2:
The Mathieu groups $M_{11}, M_{12}, M_{22}, M_{23},$ and $M_{24}$ are defined as follows:
- $M_{11}=\{\sigma\in S_{11}: \sigma(S) \in S(4,5,11) \text{ for all } S\in S(4,5,11)\}.$
- $M_{12}=\{\sigma\in S_{12}: \sigma(S) \in S(5,6,12) \text{ for all } S\in S(5,6,12)\}.$
- $M_{22}=\{\sigma\in S_{22}: \sigma(S) \in S(3,6,22) \text{ for all } S\in S(3,6,22)\}.$
- $M_{23}=\{\sigma\in S_{23}: \sigma(S) \in S(4,7,23) \text{ for all } S\in S(4,7,23)\}.$
- $M_{24}=\{\sigma\in S_{24}: \sigma(S) \in S(5,8,24) \text{ for all } S\in S(5,8,24)\}.$
[definition of group transitivity]
It follows fairly easily from the definitions that
- $M_{22}$ is $3$-transitive,
- $M_{11}$ and $M_{23}$ are $4$-transitive,
- $M_{12}$ and $M_{24}$ are $5$-transitive.
I haven't been able to see this apparently-simple proof after thinking about things for a while; it's not apparent to me that a given Steiner system should have any particular symmetries, or indeed that any of these groups are nontrivial. Any insights on how to see these transitivity properties would be welcome.
The writeup contains some more information on the construction of these Steiner systems above the quoted section, but at least in the $M_{24}$ case, nothing that provides any obvious group-theoretic symmetries, since it uses the binary lexacode construction. I would therefore assume that the reader is intended to prove that the automorphism group of a Steiner system $S(t,k,n)$ is always $t$-transitive, but this is not the case when $t=2$ (as discussed in Clapham 1976), so the proof must use $t>2$ in some crucial way. I'm at a loss as to what this proof method is, however.
(As a side note, this writeup seems to disagree with Wikipedia on whether $M_{22}$ is the automorphism group of the Steiner system, or an index-$2$ subgroup of said automorphism group. Any clarification on this front would be welcome.)