ELI5: Langrange's Theorem

Lagrange is the theorem which tells us that subgroups of finite groups have orders which divide the order of the whole group. From this theorem, you know that (for example) a subgroup from group with 24 elements must have an order which is in the set {1, 2, 3, 4, 6, 8, 12, 24}, subgroups are in some way a nice fraction of the whole group.

The proof the theorem uses the idea of a partition. One way to think of a partition of a set (or group) is to think of putting the elements of the group into "boxes". The idea of the boxes here is that there is no overlap in the partition (maybe you can think of using slices to partition a cake or pie, but the box analogy works better for me).

In the proof of Lagrange you can use to subgroup to create a special partition of the group with the property that each box in partition contains equal numbers of elements. In this case, both the number of boxes and the size of each box divides the order of the group, and, in fact, the number of elements in the group is the product of those two numbers.... imagine dividing a group of 24 elements equally into 3 boxes... then each box will have 8 elements and 24 = 3\x8.

In the special partition of the group G, one of the boxes will be a subgroup of G... if you can show the special partition does in fact equally divide the group then you will have shown, just as 3\x8 = 24, that the order of the subgroup divides the order of the group. (and that quotient is called the index of the subgroup, a sort of measure of "what fraction" the subgroup is of the whole group).

All that is background as to what the proof of Lagrange will yield, provided you know how to use the subgroup to generate the special partition of the whole group (and verify this special partition does in fact equally divide the whole group with the subgroup being one slice of the partition).

The Proof

Let H be a subgroup of a finite group G and consider the set of elements of H. Then for each element g in G, consider the set of elements given by gH, that is, for the fixed g in G, take the set of elements H and left multiply them by g. For each g in G, the size of gH is exactly equal to H (can you show this?) and for different x and y in G, either xH = yH or they have empty intersection (can you show this?).

Those two facts prove that the cosets (all those gH sets) from a partition of G with the special property that each slice of the partition is of equal size and that H is one of those slices... giving the result that the order of H divides the order of G (and the index of H in G denoted [G:H] is the quotient of their orders).

This is just an explanation of what Lagrange is trying to show and how it shows it, but it is not a proof... the real work is in showing that the cosets do form a partition of equal size slices with H being one of the slices.

/r/explainlikeimfive Thread