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February 19th, 2015, 11:32 AM   #1
msv
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Group Theory Proofs, least common multiples, and group operations HELP!

(1)
Let G be an infinite cyclic group and let m,n be integers. Prove that <a^m> intersect <a^n> = <a^d> when d is the least common multiple of m,n.

(2)
On the set G = Z x {-1, 1} = {(m,a) : m is in Z and a is in the set {-1.1}} we define the operation * by (m,a) * (n,b) = (m + an, ab). Is G a group under this operation? Is the operation commutative?
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February 19th, 2015, 12:19 PM   #2
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For the first question, an element of <a^m> is of the form (a^m)^x = a^mx for some nonnegative integer x, and an element of <a^n> is of the form (a^n)^y = a^ny. If a^mx = a^ny then mx = ny (it can't wrap around or it would be a finite cyclic group, counter to our assumption). But then mx must be divisible by n and ny must be divisible by m, and clearly mx is divisible by m and ny is divisible by y, so mx = ny must be divisible by lcm(m, n).
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