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 (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] proof of Cauchy's theorem (Proof)

Let $G$ be a finite group, and suppose $p$ is a prime divisor of $\vert G\vert$. Consider the set $X$ of all $p$-tuples $(x_1, \ldots, x_p)$ for which $x_1\cdots x_p = 1$. Note that $\vert X\vert = \vert G\vert^{p-1}$ is a multiple of $p$. There is a natural group action of the cyclic group $\mathbb{Z}/p\mathbb{Z}$ on $X$ under which $m \in \mathbb{Z}/p\mathbb{Z}$ sends the tuple $(x_1, \ldots, x_p)$ to $(x_{m+1}, \ldots, x_p, x_1, \ldots, x_m)$. By the Orbit-Stabilizer Theorem, each orbit contains exactly $1$ or $p$ tuples. Since $(1,\ldots, 1)$ has an orbit of cardinality $1$, and the orbits partition $X$, the cardinality of which is divisible by $p$, there must exist at least one other tuple $(x_1,\ldots, x_p)$ which is left fixed by every element of $\mathbb{Z}/p\mathbb{Z}$. For this tuple we have $x_1 = \ldots = x_p$, and so $x_1^p=x_1\cdots x_p=1$, and $x_1$ is therefore an element of order $p$.

References

1
James H. McKay. Another Proof of Cauchy's Group Theorem, American Math. Monthly, 66 (1959), p119.



"proof of Cauchy's theorem" is owned by yark. [ full author list (2) | owner history (1) ]
(view preamble | get metadata)

View style:


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

Cross-references: order, fixed, partition, cardinality, orbit, orbit-stabilizer theorem, tuple, cyclic group, group action, prime divisor, finite group

This is version 5 of proof of Cauchy's theorem, born on 2002-02-19, modified 2007-06-04.
Object id is 2186, canonical name is ProofOfCauchysTheorem.
Accessed 9864 times total.

Classification:
AMS MSC: 20D99 (Group theory and generalizations :: Abstract finite groups :: Miscellaneous)
 20E07 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Subgroup theorems; subgroup growth)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

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