The set of Lagrange is a mathematical theorem of group theory . In its simplest form it says that the cardinality (or order ) of each subgroup of a finite group shares its cardinality. It was named after the Italian mathematician Joseph-Louis Lagrange .
In particular, for both and also divisors of .
Proof of the theorem
For each, consider the left minor class .
It is a bijection between and , because the mapping is surjective due to the definition of a left secondary class and, according to the truncation rule, also injective. Thus all left secondary classes have the same power as the subgroup .
Since the secondary classes can be defined as equivalence classes of the equivalence relation , they provide a partition of . If one chooses a representative system of the secondary classes with the help of the axiom of choice , then one has a bijection between and through the mapping . After the definition of the index and the system of representatives, the following applies and thus one obtains
which was to be proved.
Since the order of a group element is precisely the order of the subgroup that is generated by this element, it follows from Lagrange's theorem that the order of a group element always shares the order of the group.
Finite groups, whose group order is a prime number , are cyclic and simple according to Lagrange's theorem . Since the group order is a prime number, according to Lagrange's theorem, there can only be the trivial subgroups and thus every non-neutral element already creates the whole group and there are only the trivial normal divisors.
Be a group, subgroups. Then one obtains by applying Lagrange's theorem twice
If you choose , you get Lagrange's theorem again.
Subgroups of a given order
With Lagrange's theorem one has a necessary criterion for the existence of a subgroup of a certain order for finite groups . However, the criterion is not sufficient , that is, in general, for finite groups there is not a subgroup that has this order for every divisor of the group order. The smallest group that makes this clear is the group . has elements, but no subgroup of order .
Nevertheless, there are certain groups, which for every part of the group order also have a subgroup of this order. The cyclic groups are an example . There are also sentences that guarantee the existence of subgroups of certain orders. An example of this are the Sylow sentences .
- Kurt Meyberg: Algebra - Part 1 . Hanser 1980, ISBN 3-446-13079-9 , p. 47.
- Gerd Fischer: Textbook of Algebra . Vieweg 2008, ISBN 978-3-8348-0226-2 , p. 28.