2. Let G be a group, and let H be a subgroup of G. Recall that the Normality Test says that HG if and only if ghg-¹ E H for all g E 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 E H for all g € G and all y E Y. -1 (b) Show that if G is finite and X C G is a generating set for G, then HG if and only if xhx E H for all x EX and all h E 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
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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
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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