#28 Section 10 Exercises 103 28. Let H be a subgroup of a group G such that ghg

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#28

Section 10 Exercises 103 28. Let H be a subgroup of a group G such that ghg e H for all g e G and all h e H. Show that every left coset g H is the same as the right coset Hg. 29. Let H be a subgroup of a group G. Prove that if the partition of G into left cosets of H is the same as the partition into right cosets of H, then g hg e H for all g e G and all h e H. (Note that this is the converse of Exercise 28.) 3 prove the statement or give a Let H be a subgroup ofa counterexample 30. If a H bH,then Ha = Hb.

Explanation / Answer

28) RESULT : The set of cosets of H in G partition G (i.e., every element of G belongs to some coset of H.

With this in mind, our conclusion is simple. Consider the set of left cosets of H in G. We know that 1H=H is one particular left coset. Now consider GH, the set of elements in G which are not in H. By our fact, the elements of GH belong to some coset of H in G. There are only two cosets, since the index of H in G is two. Since they are not in H, the elements of GH must belong to the second left coset of H in G. Hence, the two left cosets of H in G are therefore H and GH.

Similarly, we can observe that H1=H is a right coset in G. By the same exact reasoning as above, the only other possible right coset of H in G is GH. The two right cosets of H in G are therefore H and GH. The left and right cosets hence coincide.

Hence proved.

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