PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (191)
Orphanage
Unclass'd (1)
Unproven (496)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Low Entry average rating: No information on entry rating
[parent] $C_{mn}\cong C_m\times C_n$ when $m, n$ are relatively prime (Proof)

We show that $ C_{mn}$, gcd$ (m, n)=1$, is isomorphic to $ C_m\times C_n$, where $ C_r$ denotes the cyclic group of order $ r$ for any positive integer $ r$.

Let $ C_m=\langle x\rangle$ and $ C_n=\langle y\rangle$. Then the external direct product $ C_m\times C_n$ consists of elements $ (x^i, y^j)$, where $ 0\leq i\leq m-1$ and $ 0\leq j\leq n-1$.

Next, we show that the group $ C_m\times C_n$ is cyclic. We do so by showing that it is generated by an element, namely $ (x, y)$: if $ (x, y)$ generates $ C_m\times C_n$, then for each $ (x^i, y^j)\in C_m\times C_n$, we must have $ (x^i, y^j)=(x, y)^k$ for some $ k\in\{0, 1, 2, \ldots, mn-1\}$. Such $ k$, if exists, would satisfy

$\displaystyle k$ $\displaystyle \equiv$ $\displaystyle i\;(mod\;m)$  
$\displaystyle k$ $\displaystyle \equiv$ $\displaystyle j\;(mod\;n).$  

Indeed, by the Chinese Remainder Theorem, such $ k$ exists and is unique modulo $ mn$. (Here is where the relative primality of $ m, n$ comes into play.) Thus, $ C_m\times C_n$ is generated by $ (x, y)$, so it is cyclic.

The order of $ C_m\times C_n$ is $ mn$, so is the order of $ C_{mn}$. Since cyclic groups of the same order are isomorphic, we finally have $ C_{mn}\cong C_m\times C_n$.



"$C_{mn}\cong C_m\times C_n$ when $m, n$ are relatively prime" is owned by yesitis.
(view preamble)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: primality, Chinese remainder theorem, generates, generated by, cyclic, group, direct product, integer, positive, order, cyclic group, isomorphic, gcd

This is version 5 of $C_{mn}\cong C_m\times C_n$ when $m, n$ are relatively prime, born on 2008-04-16, modified 2008-04-17.
Object id is 10508, canonical name is C_mncongC_mtimesC_nWhenMNAreRelativelyPrime.
Accessed 260 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)