Find substructures within groups and partition elements into equal-size cosets
Within every group live smaller groups -- subgroups. The rotations inside a dihedral group form a subgroup. The even permutations inside S_n form the alternating group A_n. Subgroups partition the group into equal-size pieces called cosets, leading to Lagrange's beautiful theorem.
Select a subgroup and watch it light up on the Cayley diagram. Notice how subgroup elements cluster together.
|H| = 1, |G| = 6, index = 6
Key insight: A subgroup is a subset that is itself a group under the same operation. It must contain the identity, be closed under multiplication, and contain all inverses.
Color each coset a different color on the Cayley diagram. The cosets form a perfect partition -- every element belongs to exactly one coset, and all cosets have the same size.
All cosets have equal size (1). Normal subgroup.
Key insight: Left cosets gH = { gh : h in H }. They partition G into |G|/|H| pieces, each of size |H|. This is why subgroup order always divides group order.
For any finite group G and subgroup H, |H| divides |G|. Explore different groups and see that every subgroup order is a divisor of the group order.
Lagrange's Theorem: The order of every subgroup divides the order of the group.
| Subgroup | |H| | |G| | |G|/|H| | Divides? |
|---|---|---|---|---|
| H0 | 1 | 6 | 6 | Yes |
| H1 | 2 | 6 | 3 | Yes |
| H2 | 2 | 6 | 3 | Yes |
| H3 | 2 | 6 | 3 | Yes |
| H4 | 3 | 6 | 2 | Yes |
| H5 | 6 | 6 | 1 | Yes |
Subgroup orders as fractions of |G| = 6:
Key insight: Lagrange's theorem is a powerful constraint. A group of order 12 can only have subgroups of order 1, 2, 3, 4, 6, or 12. The converse is false: not every divisor need correspond to a subgroup.