Discrete Mathematical Structures by Pearson

Algebraic Structures 7-31

Then, H

1

∪H

2

= {0, ± 2, ± 3, ± 4, ± 6,…} is not a group. 2 + 3 = 5 does not belong to H

1

∪H

2

and thus addition is not a binary operation on H

1

∪H

2

.

7.9 COSET DECOMPOSITION

Let G be a group and H any subgroup of G. Let a be any element of G. The collection {h

1

a,

h

2

a,…, h

i

a,…}, where h

i

∈ H is defined as right coset of H in G and is denoted by Ha.

Thus,

Ha = {h

1

a, h

2

a,…,h

i

a,…}

Similarly, the collection aH = {ah

1

,ah

2

,…,ah

i

,…} is called left coset of H in G.

Since eH = He = H,

H is both a left coset and a right coset of H in G, e being the identity element of G.

If h ∈ H, then hH = H as the product of element of H by h are the elements of H in some order.

If G is a group under addition, then the cosets of H in G are defined by, ...