Journal of Scientific Research & Reports 11(1): 1-10, 2016; Article no.JSRR.26346 ISSN: 2320-0227 SCIENCEDOMAIN international www.sciencedomain.org Identification, Generating and Classification of Morphisms between Finite Groups D. Samaila 1* and M. Pius Pur 1 1Department of Mathematics, Faculty of Science and Science Education, Adamawa State University, P.O.Box 25 Mubi, Nigeria. Authors’ contributions This work was carried out in collaboration between both authors. Author DS designed the study, wrote the program and wrote the first draft of the manuscript. Author MPP managed the literature searches, analyses some relations between finite groups and author DS also analyzed and identify the morphisms between finite groups. Both authors read and approved the final manuscript. Article Information DOI: 10.9734/JSRR/2016/26346 Editor(s): (1) Narcisa C. Apreutesei, Technical University of Iasi, Romania. Reviewers: (1) Martino Garonzi, University of Brasilia, Brazil. (2) Xuanlong Ma, Beijing Normal University, China. Complete Peer review History: http://sciencedomain.org/review-history/14768 Received 12 th April 2016 Accepted 16 th May 2016 Review Article Published 24 th May 2016 ABSTRACT s Our goal in this paper is to generate functions f i from a finite group G to itself such that each f i’ are morphisms. The concept of the fundamental theorem of finite Abelian group was introduced and we proved the result that if G is any finite Abelian group, then the factor group G/H with H a subgroup + of G, is a finite Abelian group. Also, if G = Z r ⊗ Zs ⊗ Zt, then G ≅ Zr ⊗ Zs ⊗ Zt where r,s,t ∈Z . We finally conclude on identifying some homomorphisms and automorphisms on finite groups by listing all the possible maps from the group to itself with the help of GAP. Keywords: Finite group; homomorphism; isomorphism; automorphism; factor group. 1. INTRODUCTION countably many elements say x1, x2 ,..., x j in G In abstract algebra, a commutative group ( G, +) such that every x ∈ G can be uniquely = + + + is said to be finitely generated if there exists expressed in the form x n1x1 n2 x2 ... n j x j _____________________________________________________________________________________________________ *Corresponding author: E-mail: [email protected]; Samaila and Pur; JSRR, 11(1): 1-10, 2016; Article no.JSRR.26346 for some integers n1, n2, ..., nj. In this case, the (M, ·) is defined as a function ξ : G → M such that for all x,y∈G, ξ(x ∗ y) = ξ(x) ⋅ ξ(y). It can be set {x1, x2 ,..., x j } is called a generating set of deduced from this property that the the finite group G and that x , x ,..., x generate 1 2 j homomorphism ξ maps the identity element eG of the group G [1]. Again, it is obvious that every the group G to the identity element eM of the finite Abelian group is finitely generated. Thus, group M, and it also maps inverses to inverses in the family of finitely generated Abelian groups the sense that ξ(x-1) = ξ(x)-1 for all x∈G. Thus, consists of simple structure and can be one can easily say that ξ "is compatible with the completely classified. Examples of some groups group structure". The older notation for a that are finitely generated are the group of homomorphism ξ(x) is xξ. But this may be integers ( Z, +), the group of integers modulo n ambiguous as an index or a general subscript. A (Zn, +), e.t.c. Again by definition of direct sum of more recent trend is to write a group groups, any direct sum of finitely many finitely homomorphism on the right of the argument, generated Abelian groups is always a finitely omitting brackets, so that ξ(x) becomes xξ. In the generated Abelian group, and that every lattice area of abstract algebra where one considers form a finitely generated free Abelian group [2]. groups endowed with additional structure, a Some groups that are not finitely generated are homomorphism sometimes means a map which (Q, +) of rational numbers and ( Q*, ⋅) of non-zero respects not only the group structure, but also rational numbers. Also, it can be shown that the the extra structure. For example, in topological groups of real numbers with respect to addition, groups, a homomorphism is often required to be (ℜ, +) and real numbers with respect to continuous [1]. We shall now list some types of multiplication ( ℜ, ×) are also not finitely homomorphism in group theory as follows: generated [3]. Monomorphism: This is a group Now, the fundamental theorem of finitely homomorphism ξ that is injective, i.e. generated Abelian groups can be viewed in two preserves distinctness. different perspectives; The first aspect is the Primary decomposition formulation which states Epimorphism: Is a group homomorphism ξ that if an Abelian group G is finitely generated, that is surjective, i.e., the range of ξ is equal then it is isomorphic to a direct sum of primary to its co-domain. cyclic groups and infinite cyclic group where a primary cyclic group is a group whose order is a Isomorphism: This is a group power of a prime number. Thus, it can be homomorphism ξ that is both one-to-one and concluded that every finitely generated Abelian onto, i.e, bijective. Hence, the inverse of ξ is group is isomorphic to a group of the form also a group homomorphism. The groups G n ⊕ ⊕ ⊕ Z Zk ... Zk where the rank n ≥ 0, and and H are therefore said to be isomorphic if 1 u there exists an isomorphism between them the numbers k1, k2 ,..., ku are powers of not and they differ only in the notation of their necessarily distinct prime numbers. In particular, elements and are identical for all practical the group G is finite if and only if its rank n = 0. purposes. The values of n, k , k ,..., k can be uniquely 1 2 u Endomorphism: This is a group determined by G. The second perspective is the homomorphism ξ mapping a group into itself. Invariant factor decomposition. In this case, any In this case, the domain and co-domain are finitely generated Abelian group G can be written ξ n the same, and is called an endomorphism as a direct sum of the form Z ⊕ Z ⊕... ⊕ Z r1 ri of G. where r1 divides r2, which divides r3 and so on up ξ to ri. Again, the rank n and the invariant factors Automorphism: This is an endomorphism r ,..., r are uniquely determined by G, with a that is bijective (both one-to-one and onto). 1 i Hence, it is an isomorphism. unique order [4]. Obviously, the set of all the automorphisms of 2. GROUP HOMOMORPHISM a group G with functional composition as operation, forms a group, called the In group theory, given any two groups ( G, ∗) and automorphism group of G and denoted by (M, ·), a group homomorphism from ( G, ∗) to Aut( G). For example, the automorphism group of 2 Samaila and Pur; JSRR, 11(1): 1-10, 2016; Article no.JSRR.26346 the integers ( Z, +) contains only two elements, exponential map also yields a group the identity transformation and multiplication with homomorphism from the group of real numbers −1, it is isomorphic to the group Z/2 Z [3]. (ℜ,+) with respect to addition to the group of non- zero real numbers ( ℜ*, ⋅) with respect to 2.1 The Image and Kernel of multiplication. The kernel of the homomorphism Homomorphism is a singleton set {0} and the image consists of all the positive real numbers ℜ+. The kernel of any homomorphism δ:G →H is the set of all elements in the group G that are Moreover, given any two group homomorphisms mapped by φ, onto the identity element in H, i.e. δ:G → M and ϕ:M → N, their composition ϕ∘δ:G → N is also a group homomorphism. Hence, the Ker( δ) = { x ∈ G | δ (x) = eH}, class of all groups together with group homomorphisms as morphisms forms a and the image of δ is defined as category. For Abelian groups, if the groups G and M are Abelian, then the set of all Im( δ) = δ(G) = { δ(x) : for all x∈G}. homomorphisms Hom (G, M) from G to M is itself an Abelian group. Thus we can defined the sum Now, the kernel ker( δ) and the image Im( δ) of a δ + ϕ of two homomorphisms by: homomorphism δ can be interpreted as δ ϕ δ ϕ ∈ measuring how close the homomorphism is, to ( + )( x) = (x) + (x) for all x G. being an isomorphism. And by the First Isomorphism Theorem, the image of a group It can be proved that the sum δ + ϕ is again a homomorphism δ(G) is isomorphic to the quotient group homomorphism. group G/ker δ. Hence, the kernel Ker( δ) is an The addition δ + ϕ of the homomorphisms is important tool in the classification of compatible with the composition of isomorphism. φ ∈ homomorphisms in the following sense: if δ ψ ∈ η ∈ The kernel of δ is also a normal subgroup of the Hom( G, M); , Hom( M, N), and Hom( N, δ L), then we have group G and the image of is a subgroup of H. ∈ ∈ δ For given g G and x Ker( ), we have (φ + δ) ∘ ψ = ( φ ∘ ψ) + ( δ ∘ ψ) and η ∘ (φ + δ) = ( η ∘ φ) + ( η ∘ δ). δ -1 δ -1 δ δ δ -1δ δ (g xg ) = (g ) (x) (g) = (g) (x) (g) = -1 -1 -1 δ(g) eHδ(g) = δ(g) δ(g) = δ(g g) = δ(eG) = eH. Now, the composition is associative, hence, the set End( G) of all endomorphisms of an Abelian The homomorphism δ is a group Monomorphism group forms a ring called the endomorphism ring (i.e.