Factor Groups (Quotient Groups) A factor group, denoted by F(G), is a non-empty subset G of a group G that is closed under the operation of the group G. In...
Factor Groups (Quotient Groups)
A factor group, denoted by F(G), is a non-empty subset G of a group G that is closed under the operation of the group G. In simpler terms, F(G) consists of all the left cosets of G under the action of the group G.
An element a in G is said to belong to the factor group F(G) if a ∈ G and G/a ⊆ G. Here, G/a consists of all the right cosets of G that include a.
The factor group of G is a group under the operation of the group G, which means that the multiplication of elements in G is defined by the multiplication of elements in the factor group. Additionally, the identity element in G, which is the element e ∈ G that leaves every element unchanged when multiplied by any other element, is also an element of F(G).
The factor group of a normal subgroup of G is a subgroup of F(G). A normal subgroup is a subgroup that intersects the normalizer of G in G.
A normal subgroup G of G is said to be a normal subgroup of G if the factor group of G is equal to G. In other words, G is normal if every element in G is a coset leader.
The isomorphisms between factor groups are precisely the homomorphisms between groups. A homomorphism is a function that preserves the structure of the group, meaning that f(ab) = f(a)f(b) for all a, b ∈ G.
The factor groups are a powerful tool for understanding the structure of groups, and they have many applications in mathematics, physics, and computer science