Dedekind's modular law

The modular Act of Dedekind , named after Richard Dedekind is in mathematics, especially in group theory , a relationship between subgroups of a group .

formulation

Let there be subgroups of a group with . Then applies ${\displaystyle H,K,L}$ ${\displaystyle G}$ ${\displaystyle K\subset L}$

${\displaystyle K\cdot (H\cap L)=(K\cdot H)\cap L}$.

Here is the complex product , the point is omitted in the following. ${\displaystyle \cdot }$

Remember rule: does not depend on the brackets. ${\displaystyle KH\cap L}$ ${\displaystyle K\subset L}$

The proof is elementary: it is and because of it , so overall . Is the other way around , so is with , so is , again because of , so . ${\displaystyle K(H\cap L)\subset KH}$ ${\displaystyle K\subset L}$ ${\displaystyle K(H\cap L)\subset LL=L}$ ${\displaystyle K(H\cap L)\subset (KH)\cap L}$ ${\displaystyle x\in (KH)\cap L}$ ${\displaystyle x=kh}$ ${\displaystyle k\in K,h\in H}$ ${\displaystyle h=k^{-1}x\in KL=L}$ ${\displaystyle K\subset L}$ ${\displaystyle x=kh\in K(H\cap L)}$

Historical remark

This is an observation by Dedekind from 1897, he introduced the structure of an association there, which Dedekind then called a dual group because of the interchangeability of the links , and wrote with reference to logical similarities and number theoretic investigations

I will therefore call it the modular law, and every dual group in which it prevails may be called a dual group of the modular type .

Today the term dual group is used in a different sense, but one still speaks of the modular law. The module law, to which Dedekind refers in more abstract structures, reads there

${\displaystyle [\alpha +(\beta -\gamma )]-(\beta +\gamma )=[\alpha -(\beta +\gamma )]+(\beta -\gamma )}$.

There is the plus sign for the formation of the supremum in the association and the minus sign for the formation of the infimum. If you now replace the plus sign with the complex product and the minus sign with the formation of the average, which at least for normal divisors are lattice operations (see below), you get in modern notation

${\displaystyle [\alpha (\beta \cap \gamma )]\cap (\beta \gamma )=[\alpha \cap (\beta \gamma )](\beta \cap \gamma )}$.

If you keep writing and that's how it is . If, conversely , we can and choose and has as well . If you put this in the above formula and write for , the result is ${\displaystyle K=\beta \cap \gamma }$ ${\displaystyle L=\beta \gamma }$ ${\displaystyle K\leq L}$ ${\displaystyle K\leq L}$ ${\displaystyle \beta =K}$ ${\displaystyle \gamma =L}$ ${\displaystyle K=\beta \cap \gamma }$ ${\displaystyle L=\beta \gamma }$ ${\displaystyle H}$ ${\displaystyle \alpha }$

${\displaystyle [HK]\cap L=[H\cap L]K}$,

which is the formula presented above, except for the irrelevant order. The application to groups only played a subordinate role for Dedekind, he was primarily concerned with number theory .

Remarks

• Without the additional requirement , the sentence becomes wrong. ${\displaystyle K\subset L}$
• Da , is and Dedekind's formula can be written as a distributive law as follows : ${\displaystyle K\subset L}$ ${\displaystyle K=K\cap L}$

${\displaystyle (K\cap L)(H\cap L)=(KH)\cap L}$.

This also only applies under the condition . ${\displaystyle K\subset L}$

• Dedekind's formula is not the modular law in the subgroup association, because the complex product of two subgroups is generally not the subgroup produced by their union. In other words, despite Dedekind's formula, not every group is a modular group .
• If one considers the lattice of the normal divisors of a group, it is modular according to Dedekind's formula , because the complex product of two normal divisors is again a normal divisor and is equal to the smallest normal divisor that contains both.

Applications

The simple formula from Dedekind's modular law has many applications in group theory and is not infrequently used without mention. An auxiliary statement from a proof will be presented here as an example:

• If there is a maximal subgroup and a normal subgroup , there is no real normal subgroup between and . ${\displaystyle H<G}$ ${\displaystyle L\vartriangleleft G}$ ${\displaystyle H\cap L}$ ${\displaystyle L}$

To prove it, let us have a normal divisor with . Is to be shown . First of all is not included, because because then would also be what because is not the case. Hence, because of the maximality of . It follows as desired ${\displaystyle K\vartriangleleft G}$ ${\displaystyle H\cap L<K\leq L}$ ${\displaystyle K=L}$ ${\displaystyle K}$ ${\displaystyle H}$ ${\displaystyle K\leq L}$ ${\displaystyle K\leq H\cap L}$ ${\displaystyle H\cap L<K}$ ${\displaystyle G=KH}$ ${\displaystyle H}$

${\displaystyle L=G\cap L=(KH)\cap L=K(H\cap L)=K}$,

where the third equal sign applies because of Dedekind's modular law.

Individual evidence

1. ↑ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 1.3.14 (Dedekind's Modular Law)
2. ↑ ^{a b} Richard Dedekind: On the decomposition of numbers by their greatest common divisor . In: Collected mathematical works . tape 2 . Vieweg, Braunschweig 1931, 28., p. 103–147 , here p. 115 ( Module Law [accessed on March 20, 2019] Festschrift of the Technical University of Braunschweig on the occasion of the 69th Assembly of German Natural Scientists and Doctors, 8. 1–40 (1897)).
3. ↑ See also the article Pontryagin duality !
4. ^ PM Cohn: Basic Algebra - Groups, Rings and Fields , Springer-Verlag (2005), ISBN 1-85233-587-4 , page 55
5. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , proof of Theorem 5.4.2