Cauchy's theorem (group theory)

Short description: Existence of group elements of prime order

In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with x^p = e, where e is the identity element of G. 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 G divides the order of G. Cauchy's theorem implies that for any prime divisor p of the order of G, there is a subgroup of G whose order is p—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 pⁿ is the maximal power of p dividing the order of G, then G has a subgroup of order pⁿ (and using the fact that a p-group is solvable, one can show that G has subgroups of order p^r for any r less than or equal to n).

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.

Cauchy's theorem — Let G be a finite group and p be a prime. If p divides the order of G, then G has an element of order p.

Proof 1

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

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

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

[math]\displaystyle{ X = \{\,(x_1,\ldots,x_p) \in G^p : x_1x_2\cdots x_p = e\, \} }[/math]

of p-tuples of elements of G whose product (in order) gives the identity. Such a p-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 p − 1 elements can be chosen freely, so X has |G|^p−1 elements, which is divisible by p.

Now from the fact that in a group if ab = e then also ba = e, it follows that any cyclic permutation of the components of an element of X again gives an element of X. Therefore one can define an action of the cyclic group C_p of order p on X by cyclic permutations of components, in other words in which a chosen generator of C_p sends

[math]\displaystyle{ (x_1,x_2,\ldots,x_p)\mapsto(x_2,\ldots,x_p,x_1) }[/math].

As remarked, orbits in X under this action either have size 1 or size p. The former happens precisely for those tuples [math]\displaystyle{ (x,x,\ldots,x) }[/math] for which [math]\displaystyle{ x^p=e }[/math]. Counting the elements of X by orbits, and reducing modulo p, one sees that the number of elements satisfying [math]\displaystyle{ x^p=e }[/math] is divisible by p. But x = e is one such element, so there must be at least p − 1 other solutions for x, and these solutions are elements of order p. This completes the proof.

Uses

A practically immediate consequence of Cauchy's theorem is a useful characterization of finite p-groups, where p is a prime. In particular, a finite group G is a p-group (i.e. all of its elements have order p^k for some natural number k) if and only if G has order pⁿ for some natural number n. 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 G be a finite group where x² = e for all elements x of G. Then G has order 2ⁿ for some non negative integer n.

To show this, let |G| equal m. If m is 1, then G = {e} and the result is valid for [math]\displaystyle{ n=0 }[/math]. In the case of m ≥ 2, assume by contradiction that m has an odd prime factor p. Then, by Cauchy's theorem, G has an element x with order p. As [math]\displaystyle{ p\gt 2 }[/math], this implies that [math]\displaystyle{ x^2 \neq e }[/math], which conflicts with the initial hypothesis. Therefore m must be 2ⁿ.

It can be shown that G 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

An abelian simple group is either {e} or cyclic group C_p whose order is a prime number p. Let G be an abelian group, then all subgroups of G are normal subgroups. So, if G is a simple group, G has only normal subgroup that is either {e} or G. If |G| = 1, then G is {e}. It is suitable. If |G| ≥ 2, let a ∈ G is not e, the cyclic group [math]\displaystyle{ \langle a \rangle }[/math] is subgroup of G and [math]\displaystyle{ \langle a \rangle }[/math] is not {e}, then [math]\displaystyle{ G = \langle a \rangle. }[/math] Let n is the order of [math]\displaystyle{ \langle a \rangle }[/math]. If n is infinite, then

[math]\displaystyle{ G = \langle a \rangle \supsetneqq \langle a^2 \rangle \supsetneqq \{e\}. }[/math]

So in this case, it is not suitable. Then n is finite. If n is composite, n is divisible by prime q which is less than n. From Cauchy's theorem, the subgroup H will be exist whose order is q, it is not suitable. Therefore, n must be a prime number.

References

• Cauchy, Augustin-Louis (1845), "Mémoire sur les arrangements que l'on peut former avec des lettres données, et sur les permutations ou substitutions à l'aide desquelles on passe d'un arrangement à un autre", Exercises d'analyse et de physique mathématique (Paris) 3: 151–252, https://gallica.bnf.fr/ark:/12148/bpt6k96417945/f157.image 
• Cauchy, Augustin-Louis (1932), "Oeuvres complètes" (PDF), Lilliad - Université de Lille - Sciences et Technologies, second series (Paris: Gauthier-Villars) 13: 171–282, https://iris.univ-lille.fr/handle/1908/4035 
• Jacobson, Nathan (2009), Basic Algebra, Dover Books on Mathematics, I (Second ed.), Dover Publications, pp. 80, ISBN 978-0-486-47189-1 
• McKay, James H. (1959), "Another proof of Cauchy's group theorem", American Mathematical Monthly 66 (2): 119, doi:10.2307/2310010 
• Meo, M. (2004), "The mathematical life of Cauchy's group theorem", Historia Mathematica 31 (2): 196-221, doi:10.1016/S0315-0860(03)00003-X, https://www.sciencedirect.com/science/article/pii/S031508600300003X 

External links

• "Cauchy's theorem". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}. 
• "Proof of Cauchy's theorem". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}.