Group theory is the study of algebraic structures. This group theory deals with well known algebraic structures like rings, fields, and vector spaces. Group theory has influenced many parts of algebra. The two main branch of the group theory are linear algebraic groups and Lie groups. Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups.
G = A * B ->
A & B are invariant subgroups of G
G/A ≈ B, G/B ≈ A
Proof:
g = a b -> g a' g–1 = a b a' b–1 a–1 = a a' b b–1 a–1 = a a' a–1 ∑ A
A is invariant ; dido B.
G = { a B | a c- A } -> G/B ≈ A & similarly for B
Example: S3
H = { e, {123}, {321} } is invariant. Let Hi = { e, (j k) } ( i,j,k cyclic )
Then S3/H ≈ Hi but S3 ≠ H * Hi
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