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Let H be a subgroup of G, and define its normalizer as N(H) := {g G: gHg-¹ = H}. (i) Show that N(H) is a subgroup of G. (ii) Show that the subgroups of G that are conjugate to H are in one-to-one correspondence with the left cosets of N(H) in G.

Let H be a subgroup of G, and define its normalizer as N(H) := {g G: gHg-¹ = H}. (i) Show that N(H) is a subgroup of G. (ii) Show that the subgroups of G that are conjugate to H are in one-to-one correspondence with the left cosets of N(H) in G.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.5: Normal Subgroups
Problem 28E: 28. For an arbitrary subgroup of the group , the normalizer of in is the set . a. Prove...
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Let H be a subgroup of G, and define its normalizer as N(H) := {g G: gHg¹ = H}. (i) Show that N(H) is a subgroup of G. (ii) Show that the subgroups of G that are conjugate to H are in one-to-one correspondence with the left cosets of N(H) in G.
Transcribed Image Text:Let H be a subgroup of G, and define its normalizer as N(H) := {g G: gHg¹ = H}. (i) Show that N(H) is a subgroup of G. (ii) Show that the subgroups of G that are conjugate to H are in one-to-one correspondence with the left cosets of N(H) in G.
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