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# hw5 - H the other right coset is G H Thus the left and...

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Math 113 Homework # 5, selected solutions Fraleigh 10.39: Suppose that H is a subgroup of index 2 in a finite group G , so that | H | = | G | / 2. Then there are only two left cosets, and one of the left cosets is H , so since G is the disjoint union of the left cosets, the other left coset is G \ H . Since | H | = | G | / 2, there are also only two right cosets, and since one of the right cosets is
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Unformatted text preview: H , the other right coset is G \ H . Thus the left and right cosets are the same. Fraleigh 10.40: Let k be the order of a , which is also the order of the cyclic subgroup generated by a . By Lagrange’s theorem, k divides n , so we can write n = kd where d is an integer. Then a n = a kd = ( a k ) d = e d = e ....
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