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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 <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? 
February 19th, 2015, 11:19 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
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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