proof of Cauchy’s theorem
Let be a finite group, and suppose is a prime divisor
of .
Consider the set of all -tuples
for which .
Note that is a multiple
of .
There is a natural group action
of
the cyclic group
on
under which sends
the tuple
to .
By the Orbit-Stabilizer Theorem, each orbit contains exactly or tuples.
Since has an orbit of cardinality ,
and the orbits partition
,
the cardinality of which is divisible by ,
there must exist at least one other tuple
which is left fixed by every element of .
For this tuple we have ,
and so ,
and is therefore an element of order .
References
- 1 James H. McKay. Another Proof of Cauchy’s Group Theorem, American Math. Monthly, 66 (1959), p119.
Title | proof of Cauchy’s theorem |
---|---|
Canonical name | ProofOfCauchysTheorem |
Date of creation | 2013-03-22 12:23:30 |
Last modified on | 2013-03-22 12:23:30 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 8 |
Author | yark (2760) |
Entry type | Proof |
Classification | msc 20E07 |
Classification | msc 20D99 |