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An
-(sub)set is a (sub)set with
elements. For integers
, a Steiner system
is an
-set
and a set
of
-subsets having the property that any
-subset of
is contained in exactly one
-set in
. For example, if
, a Steiner system
is a set of
-sets, called hexads, with the property that any set of
elements of
is contained in (``can be completed to'') exactly one hexad. This note focuses on
. If
is a Steiner system of type
in a
-set
then the symmetric group
of
sends
to another Steiner system
of
. It is known that if
and
are any two Steiner systems of type
in
then there is a
such that
. In other words, a Steiner system of this type is unique up to relabelings. (This also implies that if one defines
to be the stabilizer of a fixed Steiner system of type
in
then any two such groups, for different Steiner systems in
, must be conjugate in
. 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