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COSETS

Def. Left and right cosets. Let H = {h1, h2, ... ,hm} be a subgroup of a group G. Then for any a ∈ G the complex product aH = {ah1,ah2, ... ,ahm} is called a left coset of H in G and the complex product Ha = {h1a, h2a, ... ,hma} 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.

The concept of the cosets of a group can be stated differently, possibly more clearly, as follows: Given a group G ={e1, e2, ... , en} and a subgroup H = {h1, h2, ... ,hm} of G, there are n left cosets eiH = {eih1,eih2, ... ,eihm}, i = 1,n, some of which may be identical. Similarly, there are n right cosets, Hei = {h1ei, h2ei, ... ,hnei}, i = 1,n, some of which may be identical.

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, ... ,ajH represent those cosets that are distinct then

G = a1H + a2H + .... + ajH

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 induces a partition of G into mutually disjoint left (or right) cosets of H.

2] Any two left (or right) 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 left (or right) cosets of H in G (called the index of H in G) is j where n = mj.

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