Basic notions in group theory
category of groups
subgroups, normal subgroups
quotient groups
group homomorphisms, kernel, image
(semi-)direct product, direct sum
types of groups
finite, infinite
discrete, continuous
multiplicative, additive
abelian, cyclic, simple, solvable

In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and As such, it is a concrete category. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving The study of this category is known as group theory. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups.

The monomorphisms in Grp are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

The category Grp is both complete and co-complete. In Mathematics, a complete category is a category in which all small limits exist The category-theoretical product in Grp is just the direct product of groups while the category-theoretical coproduct in Grp is the free product of groups. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Mathematics, one can often define a direct product of objectsalready known giving a new one In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological In Abstract algebra, the free product of groups constructs a group from two or more given ones The zero objects in Grp are the trivial groups (consisting of just an identity element). In Mathematics, a trivial group is a group consisting of a single element

The category of abelian groups, Ab, is a full subcategory of Grp. In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in Ab is an abelian category, but Grp is not. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist Indeed, Grp isn't even an additive category, because there is no natural way to define the "sum" of two group homomorphisms. In Mathematics, specifically in Category theory, an additive category is a Preadditive category C such that any finitely many objects A The identity function is an automorphism of every group, but the natural sum of this automorphism with itself would be a function which takes every element to its square. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself This is an endomorphism of a group if and only if it is abelian: (xy)² = x²y² if and only if xyxy = xxyy, if and only if yx = xy, in other words, when any two elements commute. In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself

More precisely, there is no way to define the sum of two group homomorphisms to make Grp into an additive category. The set of morphisms from the symmetric group of order three to itself, , has ten elements: an element z whose product on either side with every element of E is z (the homomorphism sending every element to the identity), three elements such that their product on one fixed side is always themself (the projections onto the three subgroups of order two), and six automorphisms. If Grp were an additive category, then this set E of ten elements would be a ring. In any ring, the zero element is singled out by the property that 0x=x0=0 for all x in the ring, and so z would have to be the zero of E. However, there are no two nonzero elements of E whose product is z, so this finite ring would have no zero divisors. A finite ring with no zero divisors is a field, but there is no field with ten elements. Since no such field exists, E is not a ring and Grp is not an additive category.

Every morphism f : G → H in Grp has a category-theoretical kernel (given by the ordinary kernel of algebra ker f = {x in G | f(x) = e}), and also a category-theoretical cokernel (given by the factor group of H by the normal closure of f(H) in H). In Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G Unlike in abelian categories, it is not true that every monomorphism in Grp is the kernel of its cokernel.

The notion of exact sequence is meaningful in Grp, and some results from the theory of abelian categories, such as the nine lemma, the five lemma, and their consequences hold true in Grp. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, the nine lemma is a statement about Commutative diagrams and Exact sequences valid in any Abelian category, as well as in the category In Mathematics, especially Homological algebra and other applications of Abelian category theory the five lemma is an important and widely used lemma The snake lemma however is not true in Grp. In Mathematics, particularly Homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the

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