Lagrange's Theorem

A big part of the hard work and the research in group theory is in finding specific types of subgroups inside of finite groups. If you have an arbitrary finite group of order n, then the number of distinct subsets is 2ⁿ. For reference, consider |G|=9. Then the number of subsets of G is 2⁹=512 and each of these is a possible subgroup of G. As you can see, this gets very large very quickly. Lagrange's Theorem, however, gives us a way to drastically cut down the number of possible subgroups of any group. You should read the proof of Lagrange's Theorem, because it makes use of some of the things we've already learned.

Theorem: Lagrange's Theorem
If G is a finite group and H is a subgroup of G, then |H| divides |G|.

Proof:

Let G be a finite group and let H be a subgroup of G. Now, consider the set of all left cosets of H in G, {gH : g∈G}. Note, this set does not necessarily form a factor group because H is not normal, but we still have the cosets. Now choose one representative from each coset such that {g₁H, g₂H,...,g_kH} is the set of distinct left cosets. Then we have that each element of G is a member of one of these cosets, or that

G = g₁H∪g₂H∪⋯∪g_kH.

We know that this union is disjoint - that is if i≠j then g_iH∩g_jH=∅. From this we get that

|G| = |g₁H|+|g₂H|+⋯+|g_kH|.

Also, we know that every coset contains the same number of elements, or |g_iH| = |H|. Finally, this gives us that |G| = k|H| so then k = |G|/|H| and |H| divides |G| as desired.

Lagrange's Theorem gives us that the order of every single subgroup must be a divisor of the order of the original group. What this does is immediately cuts down the number of possible subgroups of any finite group. Earlier I mentioned that if |G|=9 then there are 512 unique subsets of G. Lagrange's Theorem, though, tells us that of these 512 choices, the only subsets that could possibly be subgroups are the ones of order 3, which cuts the choices down to only 84. This might not seem like that big of a deal at this point, but Lagrange's Theorem is definitely the most important and most useful theorem in all of group theory.

I must make a point to emphasise what it is that Lagrange's Theorem does not say. It is easy to think that if k is some divisor of a group, G, then G has some subgroup of order k. This is not at all the case. Lagrange's Theorem only tells us a list of possible group orders and a criteria for when a subset cannot be a subgroup, it is in no way a subgroup test. It would be convenient if the converse of Lagrange's Theorem were true, but it is not. In fact, there are lots of groups that fail the converse. For example, A₄ has no subgroup of order 6.

I've written a lot of group theory proofs and I think about 90% of them use Lagrange's Theorem in one form or another - there are definitely a lot of places where we'd get stuck without it. For now, though, I'm going to give some of the immediate results of Lagrange's Theorem without proving them.

Theorem: |G/N|
Let G be a group and N⊳G. Then |G/N| = |G|/|N|.

Theorem: |a| divides |G|
In a finite group, the order of each element divides the order of the group.

Theorem: Prime Order Groups
Any group of a prime order is cyclic.

Theorem: a^|G| = e
If G is a finite group and a∈G then a^|G| = e.

Theorem: Fermat's Little Theorem
Let a be an integer and p be any prime number. Then a^p mod p = a mod p.

Theorem: Groups of order 2p
Let G be any finite group such that |G| = 2p where p is a prime number greater than 2. Then either G≈Z_2p or G≈D_p.

A few of these things seem to be very far away from Lagrange's Theorem and maybe they are, but I promise that they're all direct consequences. I didn't give the proofs because some of them are quite complicated, but I can write them if someone needs. There are a lot of proofs that start with "Suppose H is a subgroup with the property that..." and ends with "...but then the order of H does not divide the order of G hence contradicting Lagrange's Theorem." It is a very useful, very powerful tool.

References

Previous Related Post: Examples of Factor Groups

Text Reference: Gallain Chapter 7

Wikipedia: Lagrange's Theorem

Wolfram Mathworld: Lagrange's Theorem