What are Lagrange Points?
Lagrange's theorem exists in multiple subject areas, namely: Lagrange's median theorem in calculus ; four-square sum theorem in number theory; Lagrange's theorem (group theory) in group theory .
in
Content
Lagrange's four-square theorem
Lagrange's theorem is a theorem of group theory. Using cosets, it is proved that the order of a subgroup must be a reduced value of the order of a finite group.
Content of the theorem
Narrative: Let H be a finite group Subgroup of Divisible by Order.
The proof of the theorem is to apply in The left co-set. in Each left coset in is an equivalence class. will For left coset decomposition, since the number of elements of each equivalence class is equal, they are equal to Number of elements ( Yes on Left coset), so Divide by (number of elements) Order, the quotient is in The number of left cosets in is called Correct Index, recorded as .
Equivalence relation of cosets
Define a binary relationship : . The following proves that it is an equivalent relationship.
1) Reflexive: ;
2) Symmetry: ,therefore ,therefore ;
3) Transitivity: ,therefore ,therefore .
can prove, . So the left coset is made up of equivalence Determined equivalence classes.
Lagrange's theorem states that if the quotient Exists, then its order is equal to Correct Index .
The above expression is also true when it is an infinite group.
2. Inference
Obtained immediately by Lagrange's theorem: by finite groups One element Divisor group of order Order (considered by Generated cyclic groups).
3. Inverse proposition
The inverse proposition of Lagrange's theorem does not hold. Given a finite group And a divisor Integer of order , Does not necessarily have an order of Subgroup. The simplest example is a 4 degree alternating group , Its order is 12, but for a factor of 6, There are no subgroups of order 6. For the existence of such subgroups, Cauchy's theorem and Selo's theorem give a partial answer.
