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 xp = e, where e is the identity element of G. It is named after Augustin-Louis Cauchy, who discovered it in 1845.
[1][2] The theorem is a partial converse to Lagrange's theorem, which states that the order of any subgroup of a finite group G divides the order of G. In general, not every divisor of
[3] Cauchy's theorem states 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 pr for any r less than or equal to n).
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.
[4] 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. 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.
That element is a class xH for some x in G, and if m is the order of x in G, then xm = e in G gives (xH)m = eH in G/H, so p divides m; as before xm/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.
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 : CG(a)], so p divides the order of the centralizer CG(a) of a in G, which is a proper subgroup because a is not central.
Therefore one can define an action of the cyclic group Cp of order p on X by cyclic permutations of components, in other words in which a chosen generator of Cp sends As remarked, orbits in X under this action either have size 1 or size p. The former happens precisely for those tuples
Cauchy's theorem implies a rough classification of all elementary abelian groups (groups whose non-identity elements all have equal, finite order).
must be prime, since otherwise Cauchy's theorem applied to the (finite) subgroup generated by
This argument applies equally to p-groups, where every element's order is a power of