Cauchy's theorem (group theory)

Cauchy's theorem (group theory)
In mathematics, specifically group theory, Cauchy's theorem states that if is a finite group and is a prime number dividing the order of (the number of elements in ), then contains an element of order . That is, there is in such that is the smallest positive integer with = , where is the identity element of . It is named after Augustin-Louis Cauchy, who discovered it in 1845.
The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite group divides the order of . Cauchy's theorem implies that for any prime divisor of the order of , there is a subgroup of whose order is —the cyclic group generated by the element in Cauchy's theorem.
Cauchy's theorem is generalized by Sylow's first theorem, which implies that if is the maximal power of dividing the order of , then has a subgroup of order (and using the fact that a -group is solvable, one can show that has subgroups of order for any less than or equal to ).
Statement and proof
Many texts prove the theorem with the use of strong induction and the class equation, though considerably less machinery is required to prove the theorem in the abelian case. One can also invoke group actions for the proof.
Proof 1
We first prove the special case that where is abelian, and then the general case; both proofs are by induction on = ||, and have as starting case = which is trivial because any non-identity element now has order . Suppose first that is abelian. Take any non-identity element , and let be the cyclic group it generates. If divides ||, then ||/ is an element of order . If does not divide ||, then it divides the order [:] of the quotient group /, which therefore contains an element of order by the inductive hypothesis. That element is a class for some in , and if is the order of in , then = in gives = in /, so divides ; as before / is now an element of order in , completing the proof for the abelian case.
In the general case, let be the center of , which is an abelian subgroup. If divides ||, then contains an element of order by the case of abelian groups, and this element works for as well. So we may assume that does not divide the order of . Since does divide ||, and is the disjoint union of and of the conjugacy classes of non-central elements, there exists a conjugacy class of a non-central element whose size is not divisible by . But the class equation shows that size is [ :], so divides the order of the centralizer of in , which is a proper subgroup because is not central. This subgroup contains an element of order by the inductive hypothesis, and we are done.
Proof 2
This proof uses the fact that for any action of a (cyclic) group of prime order , the only possible orbit sizes are 1 and , which is immediate from the orbit stabilizer theorem.
The set that our cyclic group shall act on is the set
of -tuples of elements of whose product (in order) gives the identity. Such a -tuple is uniquely determined by all its components except the last one, as the last element must be the inverse of the product of those preceding elements. One also sees that those elements can be chosen freely, so has ||−1 elements, which is divisible by .
Now from the fact that in a group if = then also = , it follows that any cyclic permutation of the components of an element of again gives an element of . Therefore one can define an action of the cyclic group of order on by cyclic permutations of components, in other words in which a chosen generator of sends
.
As remarked, orbits in under this action either have size 1 or size . The former happens precisely for those tuples for which . Counting the elements of by orbits, and reducing modulo , one sees that the number of elements satisfying is divisible by . But = is one such element, so there must be at least other solutions for , and these solutions are elements of order . This completes the proof.
Uses
A practically immediate consequence of Cauchy's theorem is a useful characterization of finite -groups, where is a prime. In particular, a finite group is a -group (i.e. all of its elements have order for some natural number ) if and only if has order for some natural number . One may use the abelian case of Cauchy's Theorem in an inductive proof of the first of Sylow's theorems, similar to the first proof above, although there are also proofs that avoid doing this special case separately.
Example 1
Let be a finite group where for all elements of . Then has the order for some non negative integer . Let equal . In the case of is 1, then . In the case of , if has the odd prime factor , has the element where from Cauchy's theorem. It conflicts with the assumption. Therefore must be . is an abelian group, and is called an elementary abelian 2-group or Boolean group. The well-known example is Klein four-group.
Example 2
An abelian simple group is either or cyclic group whose order is a prime number . Let be an abelian group, then all subgroups of are normal subgroups. So, if is a simple group, has only normal subgroup that is either or . If , then is . It is suitable. If , let is not , the cyclic group is subgroup of and is not , then Let is the order of . If is infinite, then
So in this case, it is not suitable. Then is finite. If is composite, is divisible by prime which is less than . From Cauchy's theorem, the subgroup will be exist whose order is , it is not suitable. Therefore, must be a prime number.
Notes
References
External links
Articles containing proofs
Augustin-Louis Cauchy
Theorems about finite groups