Fixed point theorem (finite groups)
One of the numerous results in the theory of finite groups that are related to the Sylow theorems is a theorem called the fixed point theorem , which not least makes a fundamental statement about existence in this context . The fixed point theorem is based on a general formula, which not least includes the well-known class equation .
formulation
This fixed point theorem can be formulated as follows:
- A finite set and a prime number , a natural number and a finite group of the order are given .
- In doing so, he should operate on the basis of the external operation .
- Then the following statements apply:
-
- (i)
- (ii) In particular, if and are coprime , there is at least one fixed point.
General formula
The general formula mentioned above can be given as follows:
- Let there be a crowd and a group that is supposed to operate on .
- A system of representatives for the partition given by the paths is also given .
- Then the formula applies with regard to the thicknesses
-
- .
Inferences
The above fixed point theorem has a number of interesting applications.
About the center of finite p-groups
Here the fixed point theorem leads directly to the following result:
- A prime number and a finite p-group with an associated center are given .
- Then:
-
- (i) If a normal divisor does not consist of the neutral element alone, the average does not consist of the neutral element alone.
- (ii) In particular , if the finite p-group has more than one element, it has a nontrivial center .
To normal divisors of finite p-groups
The following structural statement results from the fixed point theorem:
- Every finite p-group of order ( prime, ) has a normal divisor of order .
literature
- Kurt Meyberg : Algebra. Part 1 (= mathematical basics for mathematicians, physicists and engineers ). Carl Hanser Verlag , Munich, Vienna 1975, ISBN 3-446-11965-5 ( MR0460010 ).
- Christian Karpfinger , Kurt Meyberg: Algebra: Groups - Rings - Body . 4th edition. Springer Spectrum , Berlin 2017, ISBN 978-3-662-54721-2 , doi : 10.1007 / 978-3-662-54722-9 .
- Gernot Stroth : Finite Groups. An introduction (= De Gruyter Studies ). Walter de Gruyter , Berlin 2013, ISBN 978-3-11-029157-5 .
Remarks
- ↑ With denotes the power of a set . Is a finite set , then is the number of elements contained in . With groups, this power is also called order.
- ↑ The external operation and the internal connection present in the given group are often denoted by the same symbol , namely . It is not uncommon for this symbol ( point ) to be completely suppressed. It is then as agreed .
- ↑ The subset consists of exactly the elements with for all . Such elements are called fixed points (under the group operation concerned).
- ↑ With the will number theory congruence referred.
- ↑ For a , the associated stabilizer and its index is in .
- ↑ A is a fixed point (in relation to the group operation at hand) if and / or applies.
- ↑ The summation condition may not be met by any . In this case the sum has the value as agreed .
- ↑ The fixed point theorem is obtained from the general formula using Lagrange's theorem .
- ↑ In Karpfinger / Meyberg (p. 99) you can find the general formula under the designation fixed point formula .
Individual evidence
- ↑ a b Kurt Meyberg: Algebra. Part 1. 1975, p. 65 ff., P. 67
- ^ Gernot Stroth: Finite groups. 2013, p. 5 ff.
- ^ Christian Karpfinger, Kurt Meyberg: Algebra: Groups - Rings - Body. 2017, p. 98 ff.
- ↑ a b Meyberg, op.cit., P. 67
- ↑ Stroth, op.cit., P. 5
- ↑ a b Karpfinger / Meyberg, op.cit., P. 99
- ↑ Stroth, op.cit., P. 6
- ↑ Meyberg, op.cit., P. 68
- ↑ Meyberg, op. Cit., Pp. 74-75