2. Let G be a group, and let H be a subgroup of G. Recall that the Normality Test says that H ◄ G if and only if ghg-¹ € H for all g € G and all h € H. In this problem we will show that it suffices to apply the Normality Test to a generating set. (a) Show that if YCH is a generating set for H, then H G if and only if gyg-¹ € H for all g E G and all y ≤ Y. (b) Show that if G is finite and XCG is a generating set for G, then H G if and only if xhx-¹E H for all x E X and all h € H. 1
2. Let G be a group, and let H be a subgroup of G. Recall that the Normality Test says that H ◄ G if and only if ghg-¹ € H for all g € G and all h € H. In this problem we will show that it suffices to apply the Normality Test to a generating set. (a) Show that if YCH is a generating set for H, then H G if and only if gyg-¹ € H for all g E G and all y ≤ Y. (b) Show that if G is finite and XCG is a generating set for G, then H G if and only if xhx-¹E H for all x E X and all h € H. 1
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.5: Isomorphisms
Problem 5E
Related questions
Question
![2. Let G be a group, and let H be a subgroup of G. Recall that the Normality Test
says that H ◄ G if and only if ghg-¹ € H for all g € G and all h € H. In this
problem we will show that it suffices to apply the Normality Test to a generating
set.
(a) Show that if YCH is a generating set for H, then H G if and only if
gyg-¹ € H for all g E G and all y ≤ Y.
(b) Show that if G is finite and XCG is a generating set for G, then H G if
and only if xhx-¹E H for all x E X and all h € H.
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe72e13b-12e8-4646-917c-376e7356872c%2F2ceeb5fd-83db-432d-b972-bb88a4e0141f%2Fwl88yat_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let G be a group, and let H be a subgroup of G. Recall that the Normality Test
says that H ◄ G if and only if ghg-¹ € H for all g € G and all h € H. In this
problem we will show that it suffices to apply the Normality Test to a generating
set.
(a) Show that if YCH is a generating set for H, then H G if and only if
gyg-¹ € H for all g E G and all y ≤ Y.
(b) Show that if G is finite and XCG is a generating set for G, then H G if
and only if xhx-¹E H for all x E X and all h € H.
1
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