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5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a', b' e G such that a and a' are in the same left coset of H, b and b' are in the same left coset of H, but ab and a'b' are in different left cosets of H.

5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a', b' e G such that a and a' are in the same left coset of H, b and b' are in the same left coset of H, but ab and a'b' are in different left cosets of 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
Section4.4: Cosets Of A Subgroup
Problem 12E: Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order...
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5. Let G be a group, and let H C G be a subgroup. Show that if H ‡ G, then there exist elements a, b, a', b' = G such that a and a' are in the same left coset of H, b and b' are in the same left coset of H, but ab and a'b' are in different left cosets of H.
Transcribed Image Text:5. Let G be a group, and let H C G be a subgroup. Show that if H ‡ G, then there exist elements a, b, a', b' = G such that a and a' are in the same left coset of H, b and b' are in the same left coset of H, but ab and a'b' are in different left cosets of H.
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