Cauchy's theorem (group theory)

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

statement

Cauchy's theorem says:

If a prime divides the group order of a finite group , then it contains an element of order . ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle p}$

classification

The theorem is a partial reversal of Lagrange's theorem , which says that the order of any subgroup of a finite group divides the order of : For every prime divisor of the group order, as Cauchy's theorem could also be formulated, there is (at least) one subgroup of order . ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle p}$

The theorem can also be viewed as a special case of Sylow's 1st theorem , which says that for every divisor of the group order that is a prime power, there is a subgroup of the order , i.e. a p-subgroup of . The Sylow theorems were proved by Peter Ludwig Mejdell Sylow considerably later than Cauchy's theorem in 1872 and for the inductive proof of the first theorem, the statement of Cauchy's theorem is required as the induction beginning. ${\displaystyle n}$ ${\displaystyle n=p^{r}}$ ${\displaystyle n}$ ${\displaystyle G}$

proof

The following proof is found in the Hungerford textbook and goes back to the mathematician JH McKay. Let be a finite group and a prime divisor of its group order. One considers the set of all - tuples with the property that the product is equal to the neutral element of . On this amount operates the cyclic group with elements by cyclic permutation. contains exactly elements, because with any specification of the first group elements in the tuple there is always exactly one last element, so that the tuple lies in - the inverse element of the given product. An element of is fixed by this operation of if and only if it has the same group element as entries . The tuple that contains the one element of is certainly such a fixed element , so such fixed elements exist. From the orbit formula it follows that for every orbit in , the number of its elements is a factor of the order of , i.e. of . Since is a prime number, only or can apply. The amount now decays in such webs, therefore, the number of the fixed elements (must ) is a multiple of his, as well as by assumption of divided. So there is at least one of the tuple that only the identity element of containing various fixed element in . But this fulfills the condition and therefore creates the searched subgroup with elements. ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle (g_{1},g_{2},\ldots g_{p})\in G^{p}}$ ${\displaystyle g_{1}\cdot g_{2}\cdots g_{p}=e}$ ${\displaystyle e}$ ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle C_{p}}$ ${\displaystyle M}$ ${\displaystyle |G|^{p-1}}$ ${\displaystyle p-1}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle C_{p}}$ ${\displaystyle p}$ ${\displaystyle g}$ ${\displaystyle p}$ ${\displaystyle g=e}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle C_{p}}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle n=1}$ ${\displaystyle n=p}$ ${\displaystyle M}$ ${\displaystyle n=1}$ ${\displaystyle p}$ ${\displaystyle |M|=|G|^{p-1}}$ ${\displaystyle |G|}$ ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle (g,\ldots ,g)}$ ${\displaystyle M}$ ${\displaystyle g}$ ${\displaystyle g^{p}=e}$ ${\displaystyle U}$ ${\displaystyle p}$

Inference

According to Cayley's theorem , every finite group is isomorphic to a subgroup of the symmetric group . One can now ask how large must be at least for such a (faithful!) Representation as a permutation group . If the order of the group is then one can explicitly specify a representation , but this value is only minimal in a few special cases. An element of prime power order can only operate faithfully on a set that contains at least elements. The existence of such an element can only be deduced from the order in the case alone - and that is Cauchy's special case of Sylow's first theorem. ${\displaystyle S_{n}}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle G}$ ${\displaystyle n=m}$ ${\displaystyle n=p^{r}}$ ${\displaystyle n=p^{r}}$ ${\displaystyle G}$ ${\displaystyle r=1}$

literature

• Thomas W. Hungerford: Algebra . 5th edition. Springer, New York 1989, ISBN 0-387-90518-9 , I.5, pp. 93 .
• Martin Hertweck: Script for the lecture Theory of Finite Groups . Ed .: University of Stuttgart. 2008, chap. 6 , p. 20 ( igt.uni-stuttgart.de [PDF; accessed on March 20, 2013]).
• Hans Wussing: The genesis of the abstract group concept . A contribution to the genesis of abstract group theory. VEB Deutscher Verlag der Wissenschaften, Berlin 1969 (on the history of the sentence).
• James McKay: Another proof of Cauchy's group theorem . In: American Mathematical Monthly . No. 66 , 1959, pp. 119 (proof of the theorem).

Web links

• Evandar: Cauchy's theorem . In: PlanetMath . (English)

Individual evidence

1. ↑ Wussing (1969)
2. ↑ Hertweck (2008), sentence 6.11.
3. ↑ Hungerford (1989) sentence 5.2
4. ↑ Hungerford (1989), p. 93
5. ↑ McKay (1959)