Question

2. Let G1, G2, and G3 be groups. Prove the following: a) If G1 = G2, then G2 = 61. b) If G = G2 and G2 = G3, then G =G3.

Answer 1

1. 2 @ G F G2 So there is a group isomorphism fi : G. - Gaz Since f is bifective P: G 7G, is oleo 80. daim fis a group homomorphism. het a, az E G₂ . Since f, is onto, fi (bi] = a a for some bube EGI Now f," (1992) = f ( F. (6) F, (bz)) .= T" (Pr (505) as event homotelebista homomorphism = f'(a) *,* (93) Showing that f' is a group isomorphism from Go to G, . Thus, G F G

③ Given G, I Gz and Gz = Gez So there are group isomorphisms fr : G, → G , G2 → G3 Since f, and fo ane bijectives, fopen is so. fao fi : G, G3 daim Bi Po is a group homomore bhi gsm. bet a, az EG, Then (Prof) (a, as) = & C F C2.0, =fe Ricaficar)) Since fr is a homomorphism ain since fa is a = (1,6) (16) agosto comophy om So claim is proved. Thus fao h is a group isomorphism from - Gi to Gz. Thus G E Gz.