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Let G be a finite group and H1, H2,…., Hk be subgroups of G. .... (a) Show that N H; = Hị n H2 n..n Hg < G. i=1 [Note: H1, H2,..., H are not necessarily all the subgroups of G] (b) If H; < H;, show that [G : H;] = [G : H;][H; : H;].

Let G be a finite group and H1, H2,…., Hk be subgroups of G. .... (a) Show that N H; = Hị n H2 n..n Hg < G. i=1 [Note: H1, H2,..., H are not necessarily all the subgroups of G] (b) If H; < H;, show that [G : H;] = [G : H;][H; : H;].

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.8: Some Results On Finite Abelian Groups (optional)
Problem 8E
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Let G be a finite group and H1, H2,..., Hk be subgroups of G. (a) Show that k H; = H1 N H2 N·...n Hk < G. i=1 [Note: H1, H2,..., H are not necessarily all the subgroups of G] (b) If H; < Hj, show that [G : H;] = [G : H;][H; : H;].
Transcribed Image Text:Let G be a finite group and H1, H2,..., Hk be subgroups of G. (a) Show that k H; = H1 N H2 N·...n Hk < G. i=1 [Note: H1, H2,..., H are not necessarily all the subgroups of G] (b) If H; < Hj, show that [G : H;] = [G : H;][H; : H;].
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