kvotientgruppe

A kvotientgruppe, also known as a factor group or quotient group in English, is a fundamental concept in abstract algebra. It is formed by taking a group G and a normal subgroup H within G. The elements of the kvotientgruppe are the distinct cosets of H in G.

A subgroup H of G is considered normal if for every element g in G, the left coset gH is equal to the right coset Hg. This condition ensures that the set of cosets can be endowed with a well-defined group operation.

The operation in the kvotientgruppe is defined as follows: for any two cosets aH and bH, their product is given by (aH)(bH) = (ab)H. This operation is associative because the original operation in G is associative, and it is well-defined due to the normality of H. The identity element of the kvotientgruppe is the coset eH, where e is the identity element of G. The inverse of a coset aH is (a^-1)H, where a^-1 is the inverse of a in G.

The kvotientgruppe is denoted by G/H. The order of the kvotientgruppe, |G/H|, is equal to the index of H in G, which is the number of distinct cosets of H in G. This is given by |G|/|H|, provided G is a finite group. The kvotientgruppe provides a way to "factor out" the structure of the normal subgroup H from the group G, revealing a simpler group structure.