Unformatted text preview: groups and a result of homework 3). 6. If d is a divisor of n , prove that the number of elements of order d in a cyclic group of order n is ϕ ( d ). 7. (a) Prove SL (2 , R ) ≤ GL (2 , R ). (b) Prove GL (2 , Q ) ≤ GL (2 , R ). 8. If G has no proper subgroups, prove that G is a cyclic group of order p , where p is a prime number. 9. Let G be a ﬁnite group and H ≤ G . For a ∈ G let f ( a ) be the least positive integer m such that a m ∈ H . Prove that f ( a ) | o ( a )....
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- Spring '11
- Algebra, Prime number, Abelian group, Subgroup, Cyclic group