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 . ${\displaystyle X}$ ${\displaystyle p}$ ${\displaystyle r}$ ${\displaystyle (G,\cdot )}$ ${\displaystyle |G|=p^{r}}$

In doing so, he should operate on the basis of the external operation . ${\displaystyle (G,\cdot )}$ ${\displaystyle G\times X\to X\;,\;(g,x)\mapsto g\cdot x\;,\;}$ ${\displaystyle X}$

Then the following statements apply:

(i) ${\displaystyle |\operatorname {Fix} _{G}(X)|\equiv |X|{\pmod {p}}}$

(ii) In particular, if and are coprime , there is at least one fixed point. ${\displaystyle |X|}$ ${\displaystyle p}$

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 . ${\displaystyle X}$ ${\displaystyle (G,\cdot )}$ ${\displaystyle G\times X\to X\;,\;(g,x)\mapsto g\cdot x\;,\;}$ ${\displaystyle X}$

A system of representatives for the partition given by the paths is also given . ${\displaystyle V\subseteq X}$ ${\displaystyle X}$

Then the formula applies with regard to the thicknesses

${\displaystyle |X|=|\operatorname {Fix} _{G}(X)|+\sum \limits _{x\in V\;{\text{mit}} \atop \;\ (G\colon G_{x})>1}{(G\colon G_{x})}}$.

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 . ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle \mathrm {Z} (G)}$

Then:

(i) If a normal divisor does not consist of the neutral element alone, the average does not consist of the neutral element alone. ${\displaystyle N\trianglelefteq G}$ ${\displaystyle \mathrm {Z} (G)\cap N}$

(ii) In particular , if the finite p-group has more than one element, it has a nontrivial center . ${\displaystyle G}$ ${\displaystyle \mathrm {Z} (G)}$

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 . ${\displaystyle G}$ ${\displaystyle |G|=p^{r}}$ ${\displaystyle p}$ ${\displaystyle r\in \mathbb {N} \;,\;r\geq 1}$ ${\displaystyle N\trianglelefteq G}$ ${\displaystyle |N|=p^{r-1}}$

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

1. ↑ 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. ${\displaystyle |M|}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle |M|}$ ${\displaystyle M}$
2. ↑ 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 . ${\displaystyle \cdot }$ ${\displaystyle gx=g\cdot x}$
3. ↑ The subset consists of exactly the elements with for all . Such elements are called fixed points (under the group operation concerned). ${\displaystyle \operatorname {Fix} _{G}(X)\subseteq X}$ ${\displaystyle x\in X}$ ${\displaystyle g\cdot x=x}$ ${\displaystyle g\in G}$
4. ↑ With the will number theory congruence referred. ${\displaystyle \equiv }$
5. ↑ For a , the associated stabilizer and its index is in . ${\displaystyle x\in X}$ ${\displaystyle G_{x}=\{g\in G\ |\ g\cdot x=x\}\leq G}$ ${\displaystyle (G\colon G_{x})}$ ${\displaystyle (G,\cdot )}$
6. ↑ A is a fixed point (in relation to the group operation at hand) if and / or applies. ${\displaystyle x\in X}$ ${\displaystyle G_{x}=G}$ ${\displaystyle (G\colon G_{x})=1}$
7. ↑ The summation condition may not be met by any . In this case the sum has the value as agreed . ${\displaystyle (G\colon G_{x})>1}$ ${\displaystyle x\in V}$ ${\displaystyle 0}$
8. ↑ The fixed point theorem is obtained from the general formula using Lagrange's theorem .
9. ↑ In Karpfinger / Meyberg (p. 99) you can find the general formula under the designation fixed point formula .

Individual evidence

1. ↑ ^{a b} Kurt Meyberg: Algebra. Part 1. 1975, p. 65 ff., P. 67
2. ^ Gernot Stroth: Finite groups. 2013, p. 5 ff.
3. ^ Christian Karpfinger, Kurt Meyberg: Algebra: Groups - Rings - Body. 2017, p. 98 ff.
4. ↑ ^{a b} Meyberg, op.cit., P. 67
5. ↑ Stroth, op.cit., P. 5
6. ↑ ^{a b} Karpfinger / Meyberg, op.cit., P. 99
7. ↑ Stroth, op.cit., P. 6
8. ↑ Meyberg, op.cit., P. 68
9. ↑ Meyberg, op. Cit., Pp. 74-75