Definition
The Mathieu group M₁₂ is a finite simple sporadic group of order 95 040. It can be realized as a sharply 5‑transitive permutation group on a set of 12 elements, and it is one of the five Mathieu groups discovered in the 19th century.
Overview
M₁₂ belongs to the family of Mathieu groups (M₁₁, M₁₂, M₂₂, M₂₃, M₂₄), which are the earliest known examples of sporadic simple groups. The group was first described by the French mathematician Émile Mathieu in 1861–1864 as part of his work on multiply transitive permutation groups. M₁₂ plays a central role in the classification of finite simple groups and appears in various combinatorial and geometric contexts, notably in the construction of the Steiner system S(5, 6, 12).
Etymology/Origin
The notation “M₁₂” derives from the surname of Émile Mathieu, who introduced the group while studying highly transitive permutation groups. The subscript “12” indicates the degree of the natural permutation representation, i.e., the number of points on which the group acts sharply 5‑transitively.
Characteristics
- Order: |M₁₂| = 95 040 = 2⁶·3³·5·11.
- Simplicity: M₁₂ is a non‑abelian simple group; it has no non‑trivial normal subgroups.
- Permutation representation: It acts sharply 5‑transitively on a set of 12 points, meaning that for any ordered 5‑tuple of distinct points there exists a unique group element sending it to any other ordered 5‑tuple.
- Presentations: A common presentation is
$$ M_{12}= \langle a,b \mid a^{2}=b^{3}=(ab)^{5}=(ab^{2})^{11}=1\rangle . $$ - Subgroup structure: Important maximal subgroups include M₁₁ (stabilizer of a point), PSL₂(11) (stabilizer of a pair of points), and the alternating group A₆ (stabilizer of a block of size 6 in the associated Steiner system).
- Connection to combinatorics: M₁₂ is the automorphism group of the Steiner system S(5, 6, 12), a collection of 6‑element subsets (blocks) of a 12‑element set such that every 5‑element subset is contained in exactly one block.
- Representations: Over the complex numbers, M₁₂ has irreducible characters of degrees 1, 11, 11, 16, 16, 45, 45, 55, 55, 66, 66, 77, and 77. It also possesses a 10‑dimensional faithful representation over the field 𝔽₃.
Related Topics
- Other Mathieu groups: M₁₁, M₂₂, M₂₃, M₂₄.
- Sporadic simple groups: The 26 exceptional finite simple groups not belonging to infinite families.
- Finite simple groups: The building blocks of all finite groups under the Jordan–Hölder theorem.
- Steiner systems: Combinatorial designs denoted S(t, k, v); in particular S(5, 6, 12) associated with M₁₂.
- Permutation groups: Groups acting on sets by bijections; concepts of transitivity and sharply k‑transitive actions.
- Group representations: Linear actions of groups on vector spaces, including character theory and modular representations.