Unformatted text preview: Every proper subgroup of S 4 is cyclic. Solution. False. 2.53 Let H be a subgroup of a f nite group G , and let a 1 H , . . . , a t H be a list of all the distinct cosets of H in G . Prove the following statements without using the equivalence relation on G de f ned by a ≡ b if b − 1 a ∈ H . (i) Prove that each g ∈ G lies in the coset gH , and that gH = a i H for some i . Conclude that G = a 1 H ∪ ··· ∪ a t H . Solution. Absent. (ii) If a , b ∈ G and aH ∩ bH ±= ∅ , prove that aH = bH . Conclude that if i ±= j , then a i H ∩ a j H = ∅ . Solution. Absent....
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- Fall '11
- Equivalence relation, Subgroup, Cyclic group, Solution.