The ORDER of a group

Definition 2- 9 .
Let ( H , ? ) be a subgroup of the group ( G, ? ) and a?G . The . set
a? H = { a? h | h ?H}
is called a LEFT COSET of H in G , and
H* a = { h*a | h ? H }
is called a RIGHT COSET of H in G.
Example .
In S3 if H = { r360 , l3 } , find a o H and H o a for all a ? S3 .
Notes .
e?H = H .
a? a ? H .
Theorem 2 -8 .
If (H , ? ) is a subgroup of the group ( G , ? ) , then a? H = H
if and only if a? H .
Proof .
Suppose that a ? H =H , we must prove that a ? H .
a = a?e ?a?H .
Then a?H ?
Conversely , if a ?H , then a? H ? H?
Let h? H ,then h = a ? (a-1?h )
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But a-1? h ? H?
Hence h? a?H ?
Then H ? a?H.
So a?H =H
Theorem 2 – 9
If ( H , ?) is a subgroup of the group (G , ? ), then a? H = b ? H
if and only if a-1? b?H .
Proof.
Suppose that a?H = b ?H .
If a ? h1? a? H, then ? h2 ?H ? a ?h1 = b ?h2
then a-1?b = h1? h2-1?
So a-1? b ?H?
Conversely , if a-1 ? b ?H ,
Then (a-1? b) ? H =H ?
So if h1?H ,then we can write h1 as
h1 = (a-1?b )? h2 for some h2?H
then a? h1 = b ? h2.
thus a?H = b ? H ?
Theorem 2 – 10 .
If ( H, ?) is a subgroup of the group (G , ?) then either the
Cosets a? H and b?H are disjoint or else a? H = b ?H .