CmnCm×Cn when m,n are relatively prime

We show that Cmn, gcd(m,n)=1, is isomorphicPlanetmathPlanetmathPlanetmath to Cm×Cn, where Cr denotes the cyclic groupMathworldPlanetmath of order r for any positive integer r.

Let Cm=x and Cn=y. Then the external direct product Cm×Cn consists of elements (xi,yj), where 0im-1 and 0jn-1.

Next, we show that the group Cm×Cn is cyclic. We do so by showing that it is generated by an element, namely (x,y): if (x,y) generates Cm×Cn, then for each (xi,yj)Cm×Cn, we must have (xi,yj)=(x,y)k for some k{0,1,2,,mn-1}. Such k, if exists, would satisfy

k i(modm)
k j(modn).

Indeed, by the Chinese Remainder TheoremMathworldPlanetmathPlanetmathPlanetmath, such k exists and is unique modulo mn. (Here is where the relative primality of m,n comes into play.) Thus, Cm×Cn is generated by (x,y), so it is cyclic.

The order of Cm×Cn is mn, so is the order of Cmn. Since cyclic groups of the same order are isomorphic, we finally have CmnCm×Cn.

Title CmnCm×Cn when m,n are relatively prime
Canonical name CmncongCmtimesCnWhenMNAreRelativelyPrime
Date of creation 2013-03-22 17:59:46
Last modified on 2013-03-22 17:59:46
Owner yesitis (13730)
Last modified by yesitis (13730)
Numerical id 8
Author yesitis (13730)
Entry type Proof
Classification msc 20A05