Unformatted text preview: S i has f nite index in G . Solution. Use induction on the number of distinct subgroups S i . 2.63 (i) Show that a left coset of h ( 1 2 ) i in S 3 may not be equal to a right coset of h ( 1 2 ) i in S 3 ; that is, there is α ∈ S 3 with α h ( 1 2 ) i ±= h ( 1 2 ) i α . Solution. Absent. (ii) Let G be a f nite group and let H ≤ G be a subgroup. Prove that the number of left cosets of H in G is equal to the number of right cosets of H in G . Solution. Consider aH 7→ Ha − 1 . 2.64 True or false with reasons. (i) If G and H are additive groups, then every homomorphism f : G → H satis f es f ( x + y ) = f ( x ) + f ( y ) for all x , y ∈ G . Solution. True. (ii) A function f : R → R × is a homomorphism if and only if f ( x + y ) = f ( x ) + f ( y ) for all x , y ∈ R . Solution. False....
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- Fall '11
- Normal subgroup, Equivalence relation, Coset, row sums, distinct subgroups Si