Presentation of the subgroups of Mathieu Group using Groupoid

Mathieu groups are one type of the sporadic simple groups, they turn out not to be isomorphic to any member of the infinite families of finite simple groups. Study these groups is interesting since their orders are very high. Groupoid can be used to find the presentation of the subgroups of the Mathieu groups. The idea is creating a groupoid by acting the Mathieu group on a subset of this group and then calculating the presentation of the vertex group of the groupoid which represents the presentation of the subgroup as the vertex groups are isomorphic.


Introduction
There was an attempt to complete the classification of the finite groups in particular those which are non-abelian. There are several families under the umbrella of the big family of the finite simple group. One of these families called the sporadic simple groups, it includes 26 groups, all of them are finite.Émile Léonard Mathieu (1861, 1873) introduced a special type of groups, they are multiply transitive permutation groups on n objects (n ∈ {11, 12, 22, 23, 24}). The Mathieu groups are sporadic simple groups, they are they are the first five in he family of sporadic groups. They are denoted by M 11 , M 12 , M 22 , M 23 and M 24 [7].
Groups that act on sets of nine, ten, twenty and twenty one points, respectively are denoted by M 9 , M 10 , M 20 and M 21 . These groups are subgroups of a group with large size and they are not sporadic simple groups but they can be used to construct the larger ones. This sequence can extended up to obtain M 13 , the Mathieu groupoid acting on thirteen points. Also M 21 which is not a sporadic simple group and it becomes isomorphic to projective special linear group PSL (3,4) [5]. Table 1 is showing the orders of the Mathieu group. M 12 has simple subgroup of a maximal order 660 which is isomorphic to the projective special linear group PSL2(F 11 ) over the field of 11 elements. The group M 11 is the stabilizer of a point in the group M 12 . While the group M 10 is not sporadic and it is the stabilizer of 2 points, it is an almost simple group whose commutator subgroup is the alternating group A 6 . The stabilizer of three points is PSU (3,22) "the projective special unitary group". The stabilizer of four points is Q 8 "the quaternion group". Also, M 24 has a subgroup of order 6072 (simple group) which is a maximal subgroup and it is isomorphic to PSL2(F 23 ). The stabilizers of one and two points, M 23 and M 22 also becomes simple sporadic groups. The stabilizer of three points is simple and isomorphic to PSL3(4) "the projective special linear group" [4,8].
We will try to find a presentation of a subgroup of Mathieu group by construction first a finitely presented groupoid by acting the Mathieu group on the set that generate the subgroup of the Mathieu group and then finding the presentation of the vertex group of the groupoid.
The groupoid is an algebraic structure which is a generalization of the group. It is a category in which all arrows are isomorphisms. So a group is a groupoid with one object and arrows the elements of the group.
In the context of topology, the best example of groupoid is the fundamental groupoid of a topological space in which the objects set is a set of point taken from the space and an arrow from point a to point b to be equivalence classes of paths from a to b [3]. This is generalisation of the idea of the fundamental group.
In this paper, we construct a groupoid whose objects set is the left cosets and m is an element in M (Mathieu group) and H is a subgroup of M . The morphism of the groupoid is induced by the group action, more details later.
2. Groupoids and vertex group 2.1. Groupoids, free groupoids and finitely presented groupoids A groupoid is a special type of category which is a generalization of a group.
A groupoid G is connected if for each pair of objects A and B ∈ Obj(G) there is at least one arrow w ∈ Arr(G) with the property source(w) = A and target(w) = B. The notion "free groupoid" is the corner stone of this work. Since for any free groupoid there is an underlying graph (directed graph). So let us recall the definition and required mathematical fact that help to construct such free groupoid.
consists of a set V called the set of vertices, a set E called the set of edges of Γ and two functions s, t : E → V . The vertex s(e) is the source of an edge e ∈ E. The vertex t(e) is the target of an edge e ∈ E.
) and t(f 2 (e)) = f 1 (t(e)) for all e ∈ E. Definition 2.3. The disjoint union Γ = Γ 1 Γ 2 of directed graphs Γ 1 and Γ 2 with disjoint vertex sets V (Γ 1 ) and V (Γ 2 ) and edge sets E(Γ 1 ) and E(Γ 2 ) is the directed graph with Definition 2.4. A maximal tree T of a directed graph Γ is a subgraph which includes every vertex of Γ and contains no cycle. Let Graphs denote the category whose objects are directed graphs and whose morphisms are maps of directed graphs. Let Groupoids denote the category whose objects are groupoids and whose morphisms are functors between groupoids. There is a functor which simply forgets the partial composition on a groupoid. If G is a groupoid, then the vertices of U(G) are precisely the objects of G. The directed edges of U(G) are the arrows of G.
There is a functor F : Graphs → Groupoids where for a directed graph Γ, the groupoid F(Γ) is characterized, up to isomorphism, by the following universal property.
Universal property of a free groupoid on Γ. There is a map of directed graphs ι : Γ → U(F(Γ)). For any groupoid G and any map of directed graphs f : Γ → U(G) there exists a unique groupoid morphismf : F(Γ) → G for which the following diagram commutes in the category of directed graphs. Γ We call F(Γ) the free groupoid on Γ. The existence of F(Γ) is established by an explicit construction in terms of words . When the directed graph Γ has just a single vertex we say that F(Γ) is the free group on the set E(Γ). Proof. For simplicity we denote U(G) by G for any groupoid G. Let Γ be a directed graph, and let F(Γ) and F (Γ) be free groupoids on Γ. Let ι : Γ → F(Γ) be a map, and another map ι = Γ → F (Γ). By the universal property of free groupoid there is a unique groupoid morphismῑ = F(Γ) → F (Γ) such that the following digram commutes. Now we obtain Γ By uniqueness,ῑ •ῑ = 1 F (Γ) . Similarly,ῑ •ῑ = 1 F (Γ) . Therefore, F(Γ) is isomorphic to F (Γ).
Given a discrete normal subgroupoid N in G we can form the quotient groupoid G/N which is characterized up to groupoid isomorphism by the following universal property.
Universal property of a quotient groupoid. There is a morphism of groupoids φ : G → G/N . For any groupoid Q with object set Obj(Q) = V , and for any morphism ψ : G → Q that is the identity on V and that sends each element of N to an identity element, there exists a unique morphism of groupoids ψ : G/N → Q such that the following diagram in the category of groupoids commutes.
Proof. Similar to the proof of the proposition 2.1.
Definition 2.5. We say that a set r of arrows in a discrete subgroupoid N normally generates N if any normal discrete subgroupoid of G containing r also contains the subgroupoid N .
Let G be a groupoid with vertex set V = Obj(G), and let F(Γ) be a free groupoid on a directed graph Γ = (V, x, s, t), and suppose that there is a morphism of groupoids that is the identity on objects and that is surjective on arrows. By ker φ we mean the groupoid with vertex set V and with arrows those elements r in F(Γ) mapping to an identity arrow 1 s(r) in G. The groupoid ker φ is a discrete normal subgroupoid and F(x)/ker φ is isomorphic to G. Let r be a set of elements in ker φ that normally generates ker φ. The data x | r is called a free presentation of the groupoid G.

Vertex group
Let G be a groupoid with object set Obj(G) = V . For each object (vertex) v ∈ V we let G(v, v) denote the group of arrows with source and target equal to v. We refer to G(v, v) as the vertex group or isotropy group or object group at v. The vertex group G(v, v) actually is a subgroupoid consisting of one object v and all arrows of the form v → v. Let G be a connected groupoid, we can define a homomorphism in the following sense. Let Γ be the generating graph of G, (i.e. F(Γ) = G), and let T be a maximal tree in Γ. The tree T generates a subgroupoid H of G, which called a tree of groupoid. The map θ is defined as θ(a) = v a ∈ Obj(G) θ(w) = xwy, w ∈ Arr(G), x, y ∈ H Proof. Let G be a groupoid with Obj(G) = V . Let v ∈ V and G(v, v) is the vertex group on v.
To prove that all vertex groups are isomorphic to G(v, v), let us choose any object w ∈ V , and any arrow x such that s(x) = v and t(x) = w. The map h → xhx −1 is an isomorphism from the vertex group at G(v, v) to the vertex group at G(w, w).
x ∈ x} and r = {θ(r) : r ∈ r with expressing θ(r) as a word x 1 1 x 2 2 ...x k k , x i ∈ x , i ∈ ±1} and t = {t : t edge in a maximal tree of G}.
Proof. Let x = (V, E, s, t) be a connected directed graph. Let F(x) denote the free groupoid on x. An arrow r ∈ Arr(F(x)) is said to be a loop if s(r) = t(r). Let r denote a set of loops in the groupoid F(x). Let R denote the normal subgroupoid of F(x) generated by x.
The data x | r is a presentation for the quotient groupoid Let t denote a maximal tree in the graph x. Fix some vertex v ∈ V . Then each vertex w ∈ V determines a unique simple path p(w) in the tree t with s(p(w)) = w and t(p(w)) = v. In other words, p(w) is a path in t from w to v. For each arrow a in the groupoid F(x) let us set θ(a) = p(s(a)) −1 * a * p(t(a)). Note that x is a free generating set for the free group F(v, v). here we are writing F = F(x) and letting F(v, v) denote the vertex group at v. Note that r is a subset of F(v, v). Let R(v, v) denote the normal subgroup of F(v, v) normally generated by r . We can now regard x | r as a free presentation for the finitely presented group To prove the theorem we need to see that F(v, v)/R(v, v) is isomorphic to the vertex group There is a canonical set theoretic function λ : x → G. This function induces a group homomorphism λ : The kernel of λ, by definition, consists of all loops in F(x) at v that can be written as a product of conjugates of loops in r. So clearly the kernel of λ is normally generated by r and the proof is complete.
The theorem and propositions above are implemented in GAP as a part of the package FpGd [2] available in GitHub website [1].