Adv Alegbra HW Solutions 55

Adv Alegbra HW Solutions 55 - S i has f nite index in G ....

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55 2.60 (i) Prove that the stochastic group 6( 2 , R ) , the set of all nonsingular 2 × 2 matrices whose row sums are 1, is a subgroup of GL ( 2 , R ) . Solution. The proof is a straightforward calculation: note that the formula for the inverse is needed. The students should be encour- aged to try to show that 6( 3 , R ) is a group. (ii) De f ne 6 0 ( 2 , R ) to be the set of all nonsingular doubly stochastic matrices (all row sums are 1 and all column sums are 1). Prove that 6 0 ( 2 , R ) is a subgroup of GL ( 2 , R ) . Solution. The doubly stochastic group 6 0 is a subgroup because it is the intersection of the subgroups 6 and 6 0 . 2.61 Let G be a f nite group, and let S and T be (not necessarily distinct) nonempty subsets. Prove that either G = ST or | G |≥| S |+| T | . Solution. Absent. 2.62 (i) If { S i : i I } is a family of subgroups of a group G , prove that an intersection of cosets T i I x i S i is either empty or a coset of T i I S i . Solution. Absent. (ii) ( B. H. Neumann . ) If a group G is the set-theoretic union of f nitely many cosets, G = x 1 S 1 ∪···∪ x n S n , prove that at least one of the subgroups
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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 S 3 with h ( 1 2 ) i = h ( 1 2 ) i . 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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