2.5. COSETS AND LAGRANGE’S THEOREM 125 2.5.5. What is the analogue of Proposition 2.5.3 , with left cosets replaced with right cosets? 2.5.6. Let H be a subgroup of a group G . Show that aH 7! Ha ± 1 deﬁnes a bijection between left cosets of H in G and right cosets of H in G . (The index of a subgroup was deﬁned in terms of left cosets, but this observation shows that we get the same notion using right cosets instead.) 2.5.7. For a subgroup N of a group G , prove that the following are equiv-alent: (a) N is normal. (b) Each left coset of N is also a right coset. That is, for each a 2 G , there is a b 2 G such that aN D Nb . (c) For each a 2 G , aN D Na . 2.5.8. Suppose N is a subgroup of a group G and ŒG W NŁ D 2 . Show that N is normal using the criterion of the previous exercise. 2.5.9. Show that if G is a ﬁnite group and N is a subgroup of index 2, then for elements a and b of G , the product ab is an element of N if, and only if, either both of a and b are in N or neither of a
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.