Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Lagrange's theorem (group theory)
Summary
In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order (number of elements) of every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite group , not only is an integer, but its value is the index , defined as the number of left cosets of in .
Lagrange's theorem
This variant holds even if is infinite, provided that , , and are interpreted as cardinal numbers.
The left cosets of H in G are the equivalence classes of a certain equivalence relation on G: specifically, call x and y in G equivalent if there exists h in H such that x = yh.
Therefore, the left cosets form a partition of G.
Each left coset aH has the same cardinality as H because defines a bijection (the inverse is ).
The number of left cosets is the index [G : H].
By the previous three sentences,
Lagrange's theorem can be extended to the equation of indexes between three subgroups of G.
Extension of Lagrange's theorem
Let S be a set of coset representatives for K in H,
so (disjoint union), and .
For any , left-multiplication-by-a is a bijection ,
so .
Thus each left coset of H decomposes into left cosets of K.
Since G decomposes into left cosets of H,
each of which decomposes into left cosets of K,
the total number of left cosets of K in G is .
If we take K = (e is the identity element of G), then [G : ] = G and [H : ] = H. Therefore, we can recover the original equation G = [G : H] H.
A consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integer number k with ak = e, where e is the identity element of the group) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a. If the group has n elements, it follows
This can be used to prove Fermat's little theorem and its generalization, Euler's theorem. These special cases were known long before the general theorem was proved.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.