Prove that if G is an abelian group and H is a subgroup of G

Prove that if G is an abelian group, and H is a subgroup of G, then the factor group G/H is abelian.

Solution

we know that Ian abelian group is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are applied

means a group on which group operation is commutative is called abelian group.

we know that if a and b be elements of G and G is abelian then ab = ba

as given G is abelian group and H is subgroup of G

we need to prove that factor G/H is abelian

let assume that aH and bH be element of G/H

so from the definition of the multiplication in factor groups we can say that

(aH)(bH) = (ab)H

we know that as G is abelian (ab) = (ba)

so we can say that (ab)H = (ba)H

now again from the definition of multiplication in factor groups we can say that

(ba)H = (bH)(aH)

so we have,

(aH)(bH) = (ab)H = (ba)H = (bH)(aH)

hence,

(aH)(bH) = (bH)(aH)

as aH and bH are elements of G/H and we have (aH)(bH) = (bH)(aH) we can say that G/H is abelian