A
finite group
isomorphic to one of the five groups discovered by
E. Mathieu
.
The series of Mathieu groups consists of the groups denoted by
They are representable as permutation groups (cf.
Permutation group)
on sets with 11, 12, 22, 23, and 24 elements, respectively. The groups
and
are five-fold transitive.
is realized naturally as the
stabilizer
in
of an element of the set on which
acts; similarly,
and
are stabilizers of elements of
and
,
respectively. The Mathieu groups have the respective orders
When considering a Mathieu group, one often uses (see
)
its representation as the group of automorphisms of the corresponding
Steiner system
,
i.e. of the set of
elements in which there is distinguished a system of
subsets, called blocks, consisting of
elements of the set, and such that every set of
elements is contained in one and only one block. An
automorphism of a Steiner system
is defined as a permutation of the set of
its elements which takes blocks into blocks. The list of
Mathieu groups and corresponding Steiner systems for which
they are automorphism groups is as follows:
—
;
—
;
—
;
—
;
—
.
The Mathieu groups were the first (and for over 80 years the only) known
sporadic finite simple groups
(cf. also
Sporadic simple group).