Quotientrepræsentationer

Quotientrepræsentationer, or quotient representations, are a concept in abstract algebra used to construct new algebraic structures from existing ones. Specifically, given a group G and a normal subgroup N of G, one can form the quotient group G/N. The elements of G/N are the distinct cosets of N in G. A quotient representation, in this context, refers to a way of representing these cosets, often by choosing a specific element from each coset.

The construction of a quotient group relies on the properties of normal subgroups. A subgroup N is normal if for every element g in G, the left coset gN is equal to the right coset Ng. This property ensures that the set of cosets forms a group under a well-defined operation. The operation in the quotient group G/N is defined as the product of cosets: (aN)(bN) = (ab)N.

Quotient representations are not limited to groups. The concept can be generalized to other algebraic structures like rings, modules, and vector spaces. For example, in ring theory, given a ring R and an ideal I, one can form the quotient ring R/I. The elements of R/I are the cosets of I in R, and the operations are defined analogously to group theory. The choice of representative element for each coset is a quotient representation. This process allows for the study of simpler structures by factoring out the "common" elements captured by the normal subgroup or ideal.