Cauchy's theorem (group theory)

Cauchy's theorem is a mathematical theorem of group theory, proves the existence of elements in a finite group with certain orders. The theorem is named after the French mathematician Augustin- Louis Cauchy, who proved it in 1845.

Statement

Cauchy's theorem states:

Classification

The theorem is a partial converse of the theorem of Lagrange, which states that the order of any subgroup of a finite group, the order of divides: For every prime divisor of the group order, then the Cauchy could also formulate exists (at least) a subgroup of order.

One can regard the sentence as a special case of the first Sylow theorem, which states that for every divisor of the group order, which is a prime power, are a subgroup of order, ie, a p- subgroup of. The Sylow sets were by Peter Ludwig Sylow Mejdell significantly later than Cauchy's Theorem, proved in 1872, and for the inductive proof out of the first sentence, the statement of the theorem of Cauchy as induction base is required.

Idea of ​​proof

The indicated here proof can be found in the textbook by Hungerford and goes back to the mathematician JH McKay. Let be a finite group and a prime divisor of its group order. We consider the set of all tuples with the property that the product, ie is equal to the neutral element of. On this amount the cyclic group operates with elements by cyclic permutation. contains elements precisely, since in any of the first group elements in the default tuples there is always exactly one final element so that the tuple is in - the inverse element of the given product. An element of is exactly then fixed by this operation from when it has as entries times the same group element. What is certain is the tuple that contains times the unit element of such a fixed element, so there are those fixed elements. From the railway formula now follows that for each lane in the number of its elements is a divisor of the order of, ie. Since a prime number, so it can be just or. The amount is divided now into those tracks, so the number of fixed elements () must be a multiple of, as it is also as divided by assumption of.

Conclusion

By the theorem of Cayley every finite group is isomorphic to a subgroup of the symmetric group. One can now ask how large for such a ( treue! ) View as permutation must be at least. If the order of the group, then you can explicitly specify a representation with, but this value is minimal only in a few special cases. An element of prime power order can only be true to operate on a set that contains at least elements. The existence of such an element can be inferred from the order of alone only in the case - and that is just a special case of Cauchy's first Sylowsatzes.