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                               COSETS

Def. Left and right cosets. Let H = {h₁, h₂, ... ,h_m} be a subgroup of a group G. Then for any a G the complex product aH = {ah₁,ah₂, ... ,ah_m} is called a left coset of H in G and the complex product Ha = {h₁a, h₂a, ... ,h_ma} is called a right coset of H in G. In other words, a left coset of H in G consists of the set of all products ah for all h H and any fixed element a in G. A right coset Ha of H in G consists of the set of all products ha for all h H and any fixed element a in G.

Let aH and bH be two left cosets corresponding to any two elements a, b in G. Then it can be shown that aH and bH are either identical or they have no elements in common. If a₁H, a₂H, ... ,a_jH represent those cosets that are distinct then

                        G = a₁H + a₂H + .... + a_jH

i.e. group G is partitioned by H into j mutually disjoint left cosets, which added together give group G. The number, j, of these cosets is called the index of H in G. The subgroup H is counted among its j distinct left cosets -- H corresponds to the coset eH where e is the identity element in G. Each of H’s cosets has the same number of elements as H. H is the only one of its cosets that contains the identity element e and is the only coset that is a group. If G is a finite group of order n and H is of order m the following relationship holds:

                                                n = jm

What has been stated for left cosets also applies to right cosets. However, in general, the right cosets are not equal to the left cosets. If subgroup H is normal then the left cosets are equal to the right cosets.

To repeat: Any two left (or right) cosets of a subgroup H of a group G are either identical or have no elements in common. The j distinct left (or right) cosets of H partition G into j mutually disjoint sets.

Theorems.

1] For any two cosets aH and bH of a subgroup H of a group G:

     1) aH = bH if and only if a^-1b є H

     2) If aH bH, then aH and bH are disjoint.

Thus a subgroup H defines a partition of G into mutually disjoint left-cosets of H.

2] Any two left-cosets of H in G have the same number of elements.

3] Let G be a finite group of order n and H be a subgroup of order m of G. The number of distinct right cosets of H in G (called the index of H in G) is r where n = mr.

4] (Lagrange). The order of each subgroup of a finite group G is a divisor of the order of G.

5] If G is a finite group of order n, then the order of any element a in G (i.e. the order of the cyclic subgroup generated by a) is a divisor of the order of G.

6] Every group of prime order is cyclic.

References.

  Beaumont, Ball 148-149