Question

The set G = {a ∈ Q| a≠0} is closed under the binary operation a ∗ b = ab/3 . Prove that (G, ∗) is an abelian group.

4. (10 points) The set G = {a e Qla #0} is closed under the binary operation a*b = ab 3 Prove that (G, *) is an abelian group.

Answer 1

the set = lato lato adb = 9.b فيه pread i clouriem &ab ea such that abb z gob É O 3 G is clousen onded So binanty obue -ction ( assosiativas Let gibi C EG such that. a Albfoc2 = at b c L.His at (BAC) = at bc) a bc = abe ab 3 3 (9163

this associative. (iii) Existence of undenstity! Let ас Сл and een such that so ade=a= eta ate a Oе. 3 e = 39 - 3 undentiy 3=e e (iv) Existence of unverse let taga there exist beg such that atba e b Asa a Ab = e 11e=3 ab 3 coll 3 g ab ba vo Here b. is unvense of a @ Cummtative let taibe Yuch that a Ab = ab b.g baa 3 3 Hence this is abelian Gropport