Cauchy's theorem (group theory)

Template:Short description Template:For Template:Sidebar with collapsible lists In mathematics, specifically group theory, Cauchy's theorem states that if Template:Mvar is a finite group and Template:Mvar is a prime number dividing the order of Template:Mvar (the number of elements in Template:Mvar), then Template:Mvar contains an element of order Template:Mvar. That is, there is Template:Mvar in Template:Mvar such that Template:Mvar is the smallest positive integer with Template:Mvar^{Template:Mvar} = Template:Mvar, where Template:Mvar is the identity element of Template:Mvar. It is named after Augustin-Louis Cauchy, who discovered it in 1845.Template:SfnTemplate:Sfn

The theorem is a partial converse to Lagrange's theorem, which states that the order of any subgroup of a finite group Template:Mvar divides the order of Template:Mvar. In general, not every divisor of $|G|$ arises as the order of a subgroup of $G$. Cauchy's theorem states that for any prime divisor Template:Mvar of the order of Template:Mvar, there is a subgroup of Template:Mvar whose order is Template: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 Template:Mvar^{Template:Mvar} is the maximal power of Template:Mvar dividing the order of Template:Mvar, then Template:Mvar has a subgroup of order Template:Mvar^{Template:Mvar} (and using the fact that a Template:Mvar-group is solvable, one can show that Template:Mvar has subgroups of order Template:Mvar^{Template:Mvar} for any Template:Mvar less than or equal to Template: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.Template:Sfn

Proof 1

We first prove the special case that where Template:Mvar is abelian, and then the general case; both proofs are by induction on Template:Mvar = |Template:Mvar|, and have as starting case Template:Mvar = Template:Mvar which is trivial because any non-identity element now has order Template:Mvar. Suppose first that Template:Mvar is abelian. Take any non-identity element Template:Mvar, and let Template:Mvar be the cyclic group it generates. If Template:Mvar divides |Template:Mvar|, then Template:Mvar^{|Template:Mvar|/Template:Mvar} is an element of order Template:Mvar. If Template:Mvar does not divide |Template:Mvar|, then it divides the order [[[:Template:Mvar]]:Template:Mvar] of the quotient group Template:Mvar/Template:Mvar, which therefore contains an element of order Template:Mvar by the inductive hypothesis. That element is a class Template:Mvar for some Template:Mvar in Template:Mvar, and if Template:Mvar is the order of Template:Mvar in Template:Mvar, then Template:Mvar^{Template:Mvar} = Template:Mvar in Template:Mvar gives (Template:Mvar)^{Template:Mvar} = Template:Mvar in Template:Mvar/Template:Mvar, so Template:Mvar divides Template:Mvar; as before Template:Mvar^{Template:Mvar/Template:Mvar} is now an element of order Template:Mvar in Template:Mvar, completing the proof for the abelian case.

In the general case, let Template:Mvar be the center of Template:Mvar, which is an abelian subgroup. If Template:Mvar divides |Template:Mvar|, then Template:Mvar contains an element of order Template:Mvar by the case of abelian groups, and this element works for Template:Mvar as well. So we may assume that Template:Mvar does not divide the order of Template:Mvar. Since Template:Mvar does divide |Template:Mvar|, and Template:Mvar is the disjoint union of Template:Mvar and of the conjugacy classes of non-central elements, there exists a conjugacy class of a non-central element Template:Mvar whose size is not divisible by Template:Mvar. But the class equation shows that size is [[[:Template:Mvar]] : Template:Mvar_{Template:Mvar}(Template:Mvar)], so Template:Mvar divides the order of the centralizer Template:Mvar_{Template:Mvar}(Template:Mvar) of Template:Mvar in Template:Mvar, which is a proper subgroup because Template:Mvar is not central. This subgroup contains an element of order Template: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 Template:Mvar, the only possible orbit sizes are 1 and Template:Mvar, which is immediate from the orbit stabilizer theorem.

The set that our cyclic group shall act on is the set

$X=\{(x_1,\dots ,x_p)\in G^p:x_1{x}_2\cdots x_p=e\}$

of Template:Mvar-tuples of elements of Template:Mvar whose product (in order) gives the identity. Such a Template: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 Template:Nobreak elements can be chosen freely, so Template:Mvar has |Template:Mvar|^{Template:Mvar−1} elements, which is divisible by Template:Mvar.

Now from the fact that in a group if Template:Mvar = Template:Mvar then Template:Mvar = Template:Mvar, it follows that any cyclic permutation of the components of an element of Template:Mvar again gives an element of Template:Mvar. Therefore one can define an action of the cyclic group Template:Mvar_{Template:Mvar} of order Template:Mvar on Template:Mvar by cyclic permutations of components, in other words in which a chosen generator of Template:Mvar_{Template:Mvar} sends

$(x_1,x_2,\dots ,x_p)\mapsto (x_2,\dots ,x_p,x_1)$.

As remarked, orbits in Template:Mvar under this action either have size 1 or size Template:Mvar. The former happens precisely for those tuples $(x,x,\dots ,x)$ for which $x^p=e$. Counting the elements of Template:Mvar by orbits, and dividing by Template:Mvar, one sees that the number of elements satisfying $x^p=e$ is divisible by Template:Mvar. But Template:Mvar = Template:Mvar is one such element, so there must be at least Template:Nobreak other solutions for Template:Mvar, and these solutions are elements of order Template:Mvar. This completes the proof.

Applications

Cauchy's theorem implies a rough classification of all elementary abelian groups (groups whose non-identity elements all have equal, finite order). If $G$ is such a group, and $x\in G$ has order $p$, then $p$ must be prime, since otherwise Cauchy's theorem applied to the (finite) subgroup generated by $x$ produces an element of order less than $p$. Moreover, every finite subgroup of $G$ has order a power of $p$ (including $G$ itself, if it is finite). This argument applies equally to [[p-group|Template:Mvar-groups]], where every element's order is a power of $p$ (but not necessarily every order is the same).

One may use the abelian case of Cauchy's Theorem in an inductive proofTemplate: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.

References

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External links

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