Coset

COSET

For H is a subgroup of group G, and an element a belongs to G.

Than,

aH = { ah : h   H} is called Left Coset of H in G generated by a.

Ha = { ha : h   H} is called Right Coset of H in G generated by a.

Here,

Ha and aH are both subsets of G.

If e is an identity element, than

He = H = eH  .

So in above line  H itself is a right or left coset

If group is Abelian,

Than,

ha = ah,   h   H.


FACTOR OR QUOTIENT GROUP

Factor or quotient group is a type of group which is created by the multiplication of all the left coset of a group G.

Multiplication of all left coset of G creates a new set of left coset of G.

Shown as, (aH) (bH) = (ab) H


Here ‘aH’ and ‘bH’ are left coset and ‘(ab)H ‘ is known as Factor or Quotients group.





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