### Numerical problems from GROUP

Prob. Show that the set I of all integers (…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…}
Is a group with respect to the operation of addition of integers?
Solution:

Prob. Let ({a, b}, * ) be a semigroup where a*a = b show that-
i.            a*b = b*a
Solution:

Prob. Let (A, *) be a semigroup. Show that for a, b, c ∈  A, if a*c = c*a and b*c = c*b, then (a*b)*c = c*(a*b).

Prob. Suppose (A, *) be a group, show that (A, *) is an abelian group and only if a3 * b3 = (a*b)3 for all a and b in A.
Solution:

Prob. Prove that the set Z of all integers with binary operations defined by-
a*b = a+b+1 , ∀  a, b ∈  z.  Is an abelian group.

Prob. Prove that the cube roots of unity namely ( 1, w, w2) abelian under multiplication of complex numbers.
Solution:

Prob. Prove that the set G = {0, 1, 2, 3, 4, 5} is a finite abelian group of order 6 with respect to addition modulo 6.
Solution:

Prob. Prove that the set G = {1, 2, 3, 4, 5, 6} is a finite abelian group of order 6 with respect to multiplication modulo 7.
Solution: