Breath Math

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If a and b are integers not both zero (a≠0, b≠0) then, az + bz = dz where d = gcd(a, b)
Let G be a group. A non-empty subset H of G is a subgroup of G if only if xy^-1 ϵH, for all x, y ϵH
Subgroup
Let G be a group and a, b ϵ G with O(a) = 5, a^3b = ba^3 .Prove that G is an abelian group.
Order of Group
If G is a group that (ab)^n = a^n.b^n , for all a, b ϵG and for three consecutive integers n. Then prove that G is an abelian group.
Prove that if (ab)^-1 = a^-1b^-1, for all a, b ϵG then G is abelian
Let G be a group and a^2 = e, for all a ϵG . Then prove that G is an abelian group.
If G is a group and a, b are any two elements of G then i) (ab)-1 = b-1. a-1 ii) (a-1)-1 = a iii) If a, b, c are the elements of G then ab = bc ⇒ b = c and ba = ca ⇒ b = c
Group

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