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COSETS

Def. Left and right cosets. Let H = {h_{1}, h_{2}, ... ,h_{m}} be a subgroup of a group G. Then
for any a ∈ G the complex product aH = {ah_{1},ah_{2}, ... ,ah_{m}} is called a left coset of H in G and the
complex product Ha = {h_{1}a, h_{2}a, ... ,h_{m}a} 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 ={e_{1}, e_{2}, ... , e_{n}} and a subgroup H = {h_{1}, h_{2}, ... ,h_{m}} of G, there are n left cosets
e_{i}H = {e_{i}h_{1},e_{i}h_{2}, ... ,e_{i}h_{m}}, i = 1,n, some of which may be identical. Similarly, there are n right
cosets, He_{i} = {h_{1}e_{i}, h_{2}e_{i}, ... ,h_{n}e_{i}}, 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 a_{1}H, a_{2}H, ...
,a_{j}H represent those cosets that are distinct then

G = a_{1}H + a_{2}H + .... + a_{j}H

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^{-1}b 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

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