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

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

See similar textbooks

Similar questions

the the 9.18 Suppose H and K are subgroups of a group G and x, y € G. Prove that if Hx = Ky then H=K.
12.3.7. Our friend G is a group of order 63. (a) The group G is guaranteed to have subgroups of which sizes? Why? (b) Assume that H is a subgroup of G of order 21. Prove that H < G.
Theorem 2. Let G; and G, be two groups. Let G = G,x G2 H = {(a,e,)|a e G} = G x{e;} %3D H, = {(e.b)beG} = {e}xG, and then G is an internal direct product of H and H,.
Let G be a group, let H≤G be a subgroup, and let NG be a normal subgroup. (i) Show that HnN is a normal subgroup of H. (ii) Suppose now that H is a normal subgroup of G. Show that HnN is a normal subgroup of G.
I need help with number 4
12.
Exercises 1. Let G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? a. Exercise 1 b. Exercise 2 c. Exercise 3 d. Exercise 4 e. Exercise 5 f. Exercise 6
3.4.7
2. a. If G is a group of order 175, show that G/H =Z, where H is a normal subgroup of G. b. Show that Z/nZ=Z,.
7. You have previously proved that the intersection of two subgroups of a group G is always a sub- group. For G = S3, show that the union of two subgroups may not be a subgroup by providing a counterexample.
Let H be a subgroup of G and let a, be G. If Ha = Hb, then * aH = b-H a-1H = b-1H Ha = Hb Ha-1 = Hb-1 Ha-1 = Hb-1 None of the choices
Abstract algebra Choose the correct answer