Cauchy's theorem (group theory) explained

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 a partial converse to Lagrange's theorem, which states that the order of any subgroup of a finite group divides the order of . In general, not every divisor of

|G|

arises as the order of a subgroup of

.^[1] Cauchy's theorem states 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 [{{Mvar|G}}:{{Mvar|H}}] 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 [{{Mvar|G}} : {{Mvar|C}}<sub>{{Mvar|G}}</sub>({{Mvar|a}})], 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

X=\{(x_1,\ldots,x_p)\inG^p:x_1x_{2 …}x_p=e\}

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 =, 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

(x_1,x_2,\ldots,x_p)\mapsto(x_2,\ldots,x_p,x₁₎

As remarked, orbits in under this action either have size 1 or size . The former happens precisely for those tuples

(x,x,\ldots,x)

for which

x^p=e

. Counting the elements of by orbits, and dividing by, one sees that the number of elements satisfying

x^p=e

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.

Applications

Cauchy's theorem implies a rough classification of all elementary abelian groups (groups whose non-identity elements all have equal, finite order). If

is such a group, and

x\inG

has order

, then

must be prime, since otherwise Cauchy's theorem applied to the (finite) subgroup generated by

produces an element of order less than

. Moreover, every finite subgroup of

has order a power of

(including

itself, if it is finite). This argument applies equally to -groups, where every element's order is a power of

(but not necessarily every order is the same).

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.

Notes and References

Bray . Henry G. . 1968 . A Note on CLT Groups . . 27 . 2 . 229 . Project Euclid.