Is a group homomorphism injective?

Is a group homomorphism injective?

A Group Homomorphism is Injective if and only if Monic Let f:G→G′ be a group homomorphism. We say that f is monic whenever we have fg1=fg2, where g1:K→G and g2:K→G are group homomorphisms for some group K, we have g1=g2.

Can a homomorphism be injective?

A monomorphism is an injective homomorphism, i.e. a homomorphism where different elements of G are mapped to different elements of H. A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a mapping. In this case, ker( f ) = {1G }.

Is the trivial group Simple?

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group.

Is group homomorphism Bijective?

A group homomorphism that is bijective; i.e., injective and surjective. 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. Endomorphism. A homomorphism, h: G → G; the domain and codomain are the same.

What is group homomorphism in discrete mathematics?

A group homomorphism is a map between two groups such that the group operation is preserved: for all , where the product on the left-hand side is in and on the right-hand side in .

What is homomorphism in discrete mathematics?

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: ὁμός (homos) meaning “same” and μορφή (morphe) meaning “form” or “shape”.

What does it mean if a group is simple?

A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group.

What is an example of a simple group?

The easiest examples of simple groups are the simple abelian groups. An abelian group is simple if and only if it is cyclic of prime order.

What is group Homomorphism in discrete mathematics?

What does Injective mean in math?

one-to-one function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements to distinct elements; that is, f(x1) = f(x2) implies x1 = x2. In other words, every element of the function’s codomain is the image of at most one element of its domain.