My Math Forum Group Theory Proofs, least common multiples, and group operations HELP!

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 February 19th, 2015, 10:32 AM #1 Newbie   Joined: Feb 2015 From: Alabama Posts: 3 Thanks: 0 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 intersect = 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?
 February 19th, 2015, 11:19 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms For the first question, an element of is of the form (a^m)^x = a^mx for some nonnegative integer x, and an element of 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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### the cyclic group of the least common multiple is the intersection

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