Suppose that G is a group with subgroups K ≤ H ≤ G. Suppose that K is normal in G. Let G act on G/H, the set of left cosets of H, by left multiplication. Prove that if k ∈ K, then left multiplication of G/H by k is the identity permutation on G/H.
Suppose that G is a group with subgroups K ≤ H ≤ G. Suppose that K is normal in G. Let G act on G/H, the set of left cosets of H, by left multiplication. Prove that if k ∈ K, then left multiplication of G/H by k is the identity permutation on G/H.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 13E: Let H be a subgroup of the group G. Prove that if two right cosets Ha and Hb are not disjoint, then...
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Suppose that G is a group with subgroups K ≤ H ≤ G. Suppose that K is normal in G. Let G act on G/H, the set of left cosets of H, by left multiplication. Prove that if k ∈ K, then left multiplication of G/H by k is the identity permutation on G/H.
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