QI// Let (H, *) be a normal subgroup of the group (G, *) and we define: G/H={a*H: a E G} and we define on G/H by: (a*H) O (b*H)=(a*b)*H. Prove that (G/H, ® ) is a group.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 21E
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e 0:YA
* ZAIN IQ
QI// Let (H, *) be a normal subgroup of the group (G, *) and we
define:
G/H={a*H: a € G} and we define O on G/H by:
(a*H) ® (b*H)=(a*b)*H. Prove that (G/H, ® ) is a group.
Q2// Let (H, *) be a subgroup of the group (G, *). Prove that
(a*H)N(b*H)=@ or (a*H)=(b*H) for each a, b € G.
Transcribed Image Text:e 0:YA * ZAIN IQ QI// Let (H, *) be a normal subgroup of the group (G, *) and we define: G/H={a*H: a € G} and we define O on G/H by: (a*H) ® (b*H)=(a*b)*H. Prove that (G/H, ® ) is a group. Q2// Let (H, *) be a subgroup of the group (G, *). Prove that (a*H)N(b*H)=@ or (a*H)=(b*H) for each a, b € G.
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