ProblemSet4

# ProblemSet4 - groups and a result of homework 3 6 If d is a...

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Homework 4 Due: Wednesday 2/10/10 Problems after the double bar will not be collected but it is recommended that you do them. My assumption is that you will do them. 1. Let G be an abelian group. Deﬁne H as H = { x 2 | x G } . Prove that H is a subgroup of G . 2. Let A n S n be the set consisting of all of the even permutations. Prove that A n is a subgroup of S n . 3. Let G be a ﬁnite group with subgroups H and K . If H K prove that [ G : H ] = [ G : K ][ K : H ] 4. If H and K are subgroups of G and | H | and | K | are relatively prime, prove that H K = { e } . 5. Lett G be a group of order 4. Prove that either G is cyclic or x 2 = e for all x G . (Thus we can conclude that G must be abelian as a consequence of a theorem we have about cyclic
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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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