Lagrange’s Theorem is one of the two Big theorems(along with Cayley Theorem) with actual people name on it that you can find in any basic textbook of Group Theory, see how C. Pinter describes this theorem in his book AAA.
At this point, I should describe what the theorem is, but I’m not going to do that here. Instead, I’m going to focus first on a very strange property of subgroups, and at the end, we’ll see how the theorem pops out.
Before diving in, I’m assuming that the readers know just the basics about Groups, subgroups and little bit of set therory. There are plenty of good resources out there. This theorem focuses mainly on finite Groups.We also welcome two guest groups, Z6 to help us have a concrete application of our theorem. So let’s go
subgroup is a mathematical knife!
subgroup can be used as knife to make pieces of the original Group.
Its kind of like cutting fruit with it’s own seed! or cutting an animal with it’s own bone(which sounds horrible but doable I guess!)
So let’s have a group G and take a subgroup of it H,the way we cut the group G using H is, take an element of G and multiply(operate) it with every element of H,using notation we write,
aH where a ∈ G
notice that you are multiplyng from the left side(which you also could do from the right side, but don’t bother with it now), and also notice every aH is a set it self, this cutting process has a fancy name called Coset decompostion and each of the set aH is called a Coset
If you are wondering what’s the point man? you just told us a random thing that subgroup! that is true but the magic is this cutting proccess is not random, it has two very special property and they almost tied ou
r handa about the dynamics between Group and its subgroups! This two properties which I’m going to call property 1 and 2 can also be deemed as Theorems, and in Langrange’s theorem they are lemma that we are proving.
property 1 & property 2
Property 1 : Coset decompostion (cutting with subgoup) partitions G in to disjoint subsets.
it means all of the cosets(pieces) don’t have any common elements among them and union of all them is the original Group, and this property is not trivial at all if you ask me but I’m not going to prove it here,you can try it youself as a homework, you can use the fact that
if a ∈ bH then aH ≅ bH
Property 2 : all the Cosets have the same number of element as H (alternativly all the aH and H have the same cardinality)
we can prove this using the canonical funtion we have from H to aH, for each h belong to H, we can define f(h) = ah where a remains fixed and h varies, this is valid map from f : H → aH, now if this map is injective and surjective then according to set theory H and aH have the same number of elements, can you prove this two facts?
bring them all togerther
Can you see it already? suppose H has [H] number of elements(number of elements is also known as Order of the group and it cut the Group into [M] picees (which is obviosuly a postive integer) now that every pieces(cosets) have the exact same number of element and no two pieces have anything in common the number [H][M] must be equal to [G], notice how the above two properties are absolutely important for this mutiplication fact to be true, and this is what the lagrange theorem actually or you can write [G] = [H][M]
[G]/[H] = [M]
or [G] is divisble by [H] !
now we can officialy give the statement as the order of a Group G is divislbe by order of it’s subgroup!
This is incredible because now you can use property of integer to extract infromations about Groups, this theorem is saying that if you have group of 12 elements then there can only possible subgroup of 2, 3, 4, 6 elements and no other! and also one immidate consequnce is since prime numbers dont have factors so groups with prime number of elements also can’t have any subgroups, could you guess that?
(one reminder this theroem gurantee you if subgroup of certain order exists or not, it only tells you the possibilty of existing)
chapter-5 after thought
one thing I like about math is even the simple things are ifinitly deep, the process of cutting a group using it’s subgroup shows up many places in math, like you can think of the 3-Sphere S3 as Group (multiplication of the unit quaternion) and inside this S3 you can find ordinary circle S1 which is also a Group(multiplication of unit complex numbers) that means you can use circle to cut the S3 as bundle of circles and this is part of the process known famosuly as Hopf fibration
And also since Groups of prime order are the Groups that cannot have any subgroup which Groups can have the most amount of subgroups?
well, which type of postive integer have the most number of factors?
yes! the factorial since n! is by defination divisible by every integer upto n,and the famous Symmetric Group Sn are the Group family with n! order.
If you squint your eyes enough you can see the glimps of Cayley’s theroem here, Sn are the Groups that can possibly carry the most amount of Groups possible and Cayley said they actually do carry all the finite groups there is, I also wrote another post about Cayley’s theorem.