Group Homomorphism Image And Kernel Examples The Category Of

From this property, one can deduce that h maps the identity element e_G of G to the identity element e_H of H, and it also maps inverses to inverses in the sense that h(u^-1) = h(u)^-1. Hence one can say that h "is compatible with the group structure".

Older notations for the homomorphism h(x) may be x_h, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that h(x) becomes simply x h. This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right.

In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.

1 Image and kernel

The kernel is a normal subgroup of G (in fact, h(g^-1 u g) = h(g)^-1 e_H h(g) = h(g)^-1 h(g) = e_H) and the image is a subgroup of H. The homomorphism h is injective (and called a group monomorphism) if and only if ker(h) = {e_G}.

2 Examples

3 The category of groups

If h : G -> H and k : H -> K are group homomorphisms, then so is k o h : G -> K. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a categoryCategory theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". See list of category theory topics for a breakdown of relevan.

4 Isomorphisms, endomorphisms and automorphisms

If the homomorphism h is a bijectionIn mathematics, a bijection bijective function or one-to-one correspondence is a function that is both injective ("one-to-one") and surjective ("onto"), and therefore bijections are also called one-to-one and onto . Intuitively, a bijective function creat, then one can show that its inverse is also a group homomorphism, and h is called a group isomorphismIn abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, the; in this case, the groups G and H are called isomorphic: they differ only in the notation of their elements and are identical for all practical purposes.

If h: G -> G is a group homomorphism, we call it an endomorphism of G. If furthermore it is bijective and hence an isomorphism, it is called an automorphism. The set of all automorphisms of a group G, with functional composition as operation, forms itself a group, the automorphism group of G. It is denoted by Aut(G). As an example, the automorphism group of (Z, +) contains only two elements, the identity and multiplication with -1; it is isomorphic to Z/2Z.

5 Homomorphisms of abelian groups

If G and H are abelianAbstract algebra Algebra Group theory In mathematics, an abelian group is a commutative group, i. a group G ) such that a b b a for all a and b in G''. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions fo (i.e. commutative) groups, then the set Hom(G, H) of all group homomorphisms from G to H is itself an abelian group: the sum h + k of two homomorphisms is defined by

(h + k)(u) = h(u) + k(u) for all u in G.

The commutativity of H is needed to prove that h + k is again a group homomorphism. The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in Hom(K, G), h, k are elements of Hom(G, H), and g is in Hom(H,L), then

(h + k) o f = (h o f) + (k o f) and g o (h + k) = (g o h) + (g o k).

This shows that the set End(G) of all endomorphisms of an abelian group forms a ring, the endomorphism ring of G. For example, the endomorphism ring of the abelian group consisting of the direct sumIn abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. In a sense, the direct sum of vector spaces is the "most general" vector space that contains the of two copies of Z/2Z (the Klein four-groupIn mathematics, the Klein four-group (or just Klein group or Viergruppe often symbolized by the letter V , named after Felix Klein, is a group with four elements, the smallest non- cyclic group. Its multiplication table is given by: 1abc 11abc aa1cb bbc1a) is isomorphic to the ring of 2-by-2 matricesAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number with entries in Z/2Z. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category.

Group theory

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