Question

5.- For the next question solve only (a) and (b):

(a) Let $\alpha$ = (173)(5492), $\beta$ = (23)(74)(518) $\in S_1_0$ . Write as products of disjoint cycles, and find its order. Write as a products of transpositions.

(b) Let G be a group of order p, where p is prime. Prove that G is isomorphic to $\mathbb{Z}_6$

SUBJECT: Abstract Algebra.

Answer 1

हिो 1173 (५१) (क) = 12 ___4 - (23) (74) (519) 0 () = 6 + 234567890 _d:/75 / 9463 3 2 10 1351135

I Page 4 3 4 1 1 2 17 5 3 1 5 6 7 8 9 10 1 2 9 4 6 382 1075 5 6 9 4 6 3 7 8 8 9 10 10 2 1 2 3 4 5 6 7 8 9 10 3 4 7 29 61 85 10 (137)(24)759) 1 2 3 4 5 8 3 2 7 1 6 6 7 8 4 5 9 9 10 10 P = n/ 8 3 2 2 3 7 164 s lg 1 4 5 6 7 8 9 10 and t137) (25) 459) (158)(23) (47) ('23) (5492) = (137) (24)(59) (14358792) = (15239874) 113231874) Elis) (1-7 (13) (19) (78) (17) (19) Every group of order p (p & prime) is Roof Let G be a group of onder p. - le o(G) = p. het etg EG (g is any non-identity element)

Aniket Date Page het H be a subgroup of G such that HD <g> Phone o Then by Lagrange's Theorem OCH lo CG) = Since på prime we have OCH)=1 or o(H) =p As, H contains e and g, o(H)>I .. om o (H) = þ » oG) .. Honee G is a cyclic group of order p. However, since upto Isomorphism there i i a unique cyclic group of every order .. Hence G is is omorphic to ze G ^ Zp