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An $m$-(sub)set is a (sub)set with $m$ elements. For integers $k<m<n$, a Steiner system $S(k,m,n)$ is an $n$-set $X$ and a set $S$ of $m$-subsets having the property that any $k$-subset of $X$ is contained in exactly one $m$-set in $S$. For example, if $X=\{1,2,...,12\}$, a Steiner system $S(5,6,12)$ is a set of $6$-sets, called hexads, with the property that any set of $5$ elements of $X$ is contained in (``can be completed to'') exactly one hexad. This note focuses on $S(5,6,12)$. If $S$ is a Steiner system of type $(5,6,12)$ in a $12$-set $X$ then the symmetric group $S_X$ of $X$ sends $S$ to another Steiner system $\sigma(S)$ of $X$. It is known that if $S$ and $S'$ are any two Steiner systems of type $(5,6,12)$ in $X$ then there is a $\sigma\in S_X$ such that $S'=\sigma(S)$. In other words, a Steiner system of this type is unique up to relabelings. (This also implies that if one defines $M_{12}$ to be the stabilizer of a fixed Steiner system of type $(5,6,12)$ in $X=\{1,2,...,12\}$ then any two such groups, for different Steiner systems in $X$, must be conjugate in $S_X$. In particular, such a definition is well-defined up to isomorphism.)
Next: Curtis' kitten Up: MINIMOGs and Mathematical blackjack Previous: MINIMOGs and Mathematical blackjack
David Joyner
2000-05-29