Group homomorphism

Group homomorphism . Function established between groups to preserve their structure.

Summary

[ hide ]

• 1 Definition
• 2 Theorem
• 3 Core of a homomorphism
• 4 Types of homomorphisms
• 5 Bibliography
• 6 Sources

Definition

If for two groups B and C and one application

f: B → C. It is true that:

f (x + y) = f (x) + f (y) (for all x, y).

It means that the image of element x + y of set B is the same as the image of element x + the image of element y.

We must bear in mind that the + sign does not represent the addition operation, but the operation that has been defined in the set so that it has a group structure.

Theorem

Let B and C be two groups and f: B → C a homomorphism. It is true that:

1. if 1 _Band 1 _Care the identities of B and C, respectively, then f (1 _B ) = 1 _C ; The neutral element of the domain corresponds to the neutral element of the domain, according to a homomorphism.
2. if x is in B then f (x ^{– 1}) = f (x) ^{– 1}.. The image of the inverse is equal to the inverse of the image.

Core of a homomorphism

Let B and C be two groups and let f be a homomorphism between them. The nucleus of f is defined as the set .

Ker f = {x is in B / f (x) = 1 _C , where 1 _C is the identity of C. The nucleus of a homomorphism is the set of all the elements of the domain whose image is equal to the neutral element of the codomain.

Example if f is the homomorphism of the additive group Z of the integers in the group Z ₅ of the congruence residues modulo 5, formed by {0,1,2, 3,4}, Kerf = {x integer / x = multiple of 5}

Kerf is a normal subgroup; in the example, multiples of 5, with addition, form a normal subgroup of Z.

Types of homomorphisms

• A suprajective homomorphism is called an epimorphism .
• An injection homomorphism is called a monomorphism .
• A bijective homomorphism whose inverse is also a homomorphism is called isomorphism . Two isomorphic objects are totally indistinguishable as far as the structure in question is concerned.
• A homomorphism of a set to itself is called endomorphism . If it is also an isomorphism it is called automorphism