külggruppe

Külggruppe, in group theory, refers to a construct arising from a group paired with one of its subgroups. Let G be a group and H ≤ G. For any g in G, the left coset gH is the set {gh | h ∈ H}, and the right coset Hg is {hg | h ∈ H}. These cosets are the basic building blocks used to study the structure of G relative to H.

Cosets partition G: every element of G lies in exactly one left coset of H (and exactly one right coset). All left cosets have the same size as H, and two left cosets are either identical or disjoint. The number of distinct left cosets is called the index of H in G, denoted [G:H]. If G is finite, then |G| = [G:H] · |H|.

If H is a normal subgroup of G, the set of left cosets G/H acquires a natural group structure with the operation (gH)(kH) = (gk)H, making G/H the quotient group. In general, without normality, the cosets do not form a group, though they remain a useful organizing principle.

Example: in the additive group Z of integers, the subgroup nZ consists of all multiples of n. The left cosets are the congruence classes modulo n, of which there are n. The quotient Z/nZ is a cyclic group of order n.

Properties and applications: two elements g, g' lie in the same left coset of H iff g^{-1}g' ∈ H; coset concepts underpin many counting arguments (Lagrange’s theorem), permutation representations, and the construction of quotient groups. Külggruppide framework provides a way to analyze how subgroups sit inside larger groups.