Английская Википедия:Cauchy's theorem (group theory)
Шаблон:Short description Шаблон:For Шаблон:Group theory sidebar In mathematics, specifically group theory, Cauchy's theorem states that if Шаблон:Mvar is a finite group and Шаблон:Mvar is a prime number dividing the order of Шаблон:Mvar (the number of elements in Шаблон:Mvar), then Шаблон:Mvar contains an element of order Шаблон:Mvar. That is, there is Шаблон:Mvar in Шаблон:Mvar such that Шаблон:Mvar is the smallest positive integer with Шаблон:MvarШаблон:Mvar = Шаблон:Mvar, where Шаблон:Mvar is the identity element of Шаблон:Mvar. It is named after Augustin-Louis Cauchy, who discovered it in 1845.Шаблон:SfnШаблон:Sfn
The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite group Шаблон:Mvar divides the order of Шаблон:Mvar. Cauchy's theorem implies that for any prime divisor Шаблон:Mvar of the order of Шаблон:Mvar, there is a subgroup of Шаблон:Mvar whose order is Шаблон:Mvar—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 Шаблон:MvarШаблон:Mvar is the maximal power of Шаблон:Mvar dividing the order of Шаблон:Mvar, then Шаблон:Mvar has a subgroup of order Шаблон:MvarШаблон:Mvar (and using the fact that a Шаблон:Mvar-group is solvable, one can show that Шаблон:Mvar has subgroups of order Шаблон:MvarШаблон:Mvar for any Шаблон:Mvar less than or equal to Шаблон:Mvar).
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.Шаблон:Sfn
Proof 1
We first prove the special case that where Шаблон:Mvar is abelian, and then the general case; both proofs are by induction on Шаблон:Mvar = |Шаблон:Mvar|, and have as starting case Шаблон:Mvar = Шаблон:Mvar which is trivial because any non-identity element now has order Шаблон:Mvar. Suppose first that Шаблон:Mvar is abelian. Take any non-identity element Шаблон:Mvar, and let Шаблон:Mvar be the cyclic group it generates. If Шаблон:Mvar divides |Шаблон:Mvar|, then Шаблон:Mvar|Шаблон:Mvar|/Шаблон:Mvar is an element of order Шаблон:Mvar. If Шаблон:Mvar does not divide |Шаблон:Mvar|, then it divides the order [[[:Шаблон:Mvar]]:Шаблон:Mvar] of the quotient group Шаблон:Mvar/Шаблон:Mvar, which therefore contains an element of order Шаблон:Mvar by the inductive hypothesis. That element is a class Шаблон:Mvar for some Шаблон:Mvar in Шаблон:Mvar, and if Шаблон:Mvar is the order of Шаблон:Mvar in Шаблон:Mvar, then Шаблон:MvarШаблон:Mvar = Шаблон:Mvar in Шаблон:Mvar gives (Шаблон:Mvar)Шаблон:Mvar = Шаблон:Mvar in Шаблон:Mvar/Шаблон:Mvar, so Шаблон:Mvar divides Шаблон:Mvar; as before Шаблон:MvarШаблон:Mvar/Шаблон:Mvar is now an element of order Шаблон:Mvar in Шаблон:Mvar, completing the proof for the abelian case.
In the general case, let Шаблон:Mvar be the center of Шаблон:Mvar, which is an abelian subgroup. If Шаблон:Mvar divides |Шаблон:Mvar|, then Шаблон:Mvar contains an element of order Шаблон:Mvar by the case of abelian groups, and this element works for Шаблон:Mvar as well. So we may assume that Шаблон:Mvar does not divide the order of Шаблон:Mvar. Since Шаблон:Mvar does divide |Шаблон:Mvar|, and Шаблон:Mvar is the disjoint union of Шаблон:Mvar and of the conjugacy classes of non-central elements, there exists a conjugacy class of a non-central element Шаблон:Mvar whose size is not divisible by Шаблон:Mvar. But the class equation shows that size is [[[:Шаблон:Mvar]] : Шаблон:MvarШаблон:Mvar(Шаблон:Mvar)], so Шаблон:Mvar divides the order of the centralizer Шаблон:MvarШаблон:Mvar(Шаблон:Mvar) of Шаблон:Mvar in Шаблон:Mvar, which is a proper subgroup because Шаблон:Mvar is not central. This subgroup contains an element of order Шаблон:Mvar 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 Шаблон:Mvar, the only possible orbit sizes are 1 and Шаблон:Mvar, which is immediate from the orbit stabilizer theorem.
The set that our cyclic group shall act on is the set
- <math> X = \{\,(x_1,\ldots,x_p) \in G^p : x_1x_2\cdots x_p = e\, \} </math>
of Шаблон:Mvar-tuples of elements of Шаблон:Mvar whose product (in order) gives the identity. Such a Шаблон:Mvar-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 Шаблон:Nobreak elements can be chosen freely, so Шаблон:Mvar has |Шаблон:Mvar|Шаблон:Mvar−1 elements, which is divisible by Шаблон:Mvar.
Now from the fact that in a group if Шаблон:Mvar = Шаблон:Mvar then also Шаблон:Mvar = Шаблон:Mvar, it follows that any cyclic permutation of the components of an element of Шаблон:Mvar again gives an element of Шаблон:Mvar. Therefore one can define an action of the cyclic group Шаблон:MvarШаблон:Mvar of order Шаблон:Mvar on Шаблон:Mvar by cyclic permutations of components, in other words in which a chosen generator of Шаблон:MvarШаблон:Mvar sends
- <math>(x_1,x_2,\ldots,x_p)\mapsto(x_2,\ldots,x_p,x_1)</math>.
As remarked, orbits in Шаблон:Mvar under this action either have size 1 or size Шаблон:Mvar. The former happens precisely for those tuples <math>(x,x,\ldots,x)</math> for which <math>x^p=e</math>. Counting the elements of Шаблон:Mvar by orbits, and reducing modulo Шаблон:Mvar, one sees that the number of elements satisfying <math>x^p=e</math> is divisible by Шаблон:Mvar. But Шаблон:Mvar = Шаблон:Mvar is one such element, so there must be at least Шаблон:Nobreak other solutions for Шаблон:Mvar, and these solutions are elements of order Шаблон:Mvar. This completes the proof.
Uses
A practically immediate consequence of Cauchy's theorem is a useful characterization of finite [[p-group|Шаблон:Mvar-groups]], where Шаблон:Mvar is a prime. In particular, a finite group Шаблон:Mvar is a Шаблон:Mvar-group (i.e. all of its elements have order Шаблон:MvarШаблон:Mvar for some natural number Шаблон:Mvar) if and only if Шаблон:Mvar has order Шаблон:MvarШаблон:Mvar for some natural number Шаблон:Mvar. One may use the abelian case of Cauchy's Theorem in an inductive proofШаблон:Sfn 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
Шаблон:Unreliable sources Let Шаблон:Mvar be a finite group where Шаблон:Math for all elements Шаблон:Mvar of Шаблон:Mvar. Then Шаблон:Mvar has order Шаблон:Math for some non negative integer Шаблон:Mvar.
To show this, let Шаблон:Math equal Шаблон:Mvar. If Шаблон:Mvar is 1, then Шаблон:Math and the result is valid for <math>n=0</math>. In the case of Шаблон:Math, assume by contradiction that Шаблон:Mvar has an odd prime factor Шаблон:Mvar. Then, by Cauchy's theorem, Шаблон:Mvar has an element Шаблон:Mvar with order Шаблон:Mvar. As <math>p>2</math>, this implies that <math>x^2 \neq e</math>, which conflicts with the initial hypothesis. Therefore Шаблон:Mvar must be Шаблон:Math.[1]
It can be shown that Шаблон:Mvar is also an abelian group. A group that satisfies the property of this example is called an elementary abelian 2-group or Boolean group. The well-known example is Klein four-group.
Example 2
Шаблон:Unreferenced-section An abelian simple group is either Шаблон:Math or cyclic group Шаблон:Mvar whose order is a prime number Шаблон:Mvar. Let Шаблон:Mvar be an abelian group, then all subgroups of Шаблон:Mvar are normal subgroups. So, if Шаблон:Mvar is a simple group, Шаблон:Mvar has only normal subgroup that is either Шаблон:Math or Шаблон:Mvar. If Шаблон:Math, then Шаблон:Mvar is Шаблон:Math. It is suitable. If Шаблон:Math, let Шаблон:Math is not Шаблон:Mvar, the cyclic group <math>\langle a \rangle</math> is subgroup of Шаблон:Mvar and <math>\langle a \rangle</math> is not Шаблон:Math, then <math>G = \langle a \rangle.</math> Let Шаблон:Mvar is the order of <math>\langle a \rangle</math>. If Шаблон:Mvar is infinite, then
- <math>G = \langle a \rangle \supsetneqq \langle a^2 \rangle \supsetneqq \{e\}.</math>
So in this case, it is not suitable. Then Шаблон:Mvar is finite. If Шаблон:Mvar is composite, Шаблон:Mvar is divisible by prime Шаблон:Mvar which is less than Шаблон:Mvar. From Cauchy's theorem, the subgroup Шаблон:Mvar will be exist whose order is Шаблон:Mvar, it is not suitable. Therefore, Шаблон:Mvar must be a prime number.
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