Subgroups & Cosets

Find substructures within groups and partition elements into equal-size cosets

Subgroups & Cosets

Inside most groups live smaller groups. Shift copies of one around and they tile the whole group in equal pieces.

Subgroup Highlighter

Choose a group and a subgroup, and watch its elements light up on the Cayley diagram.

|H| = 1, |G| = 6, index = 6

Go deeper:Subgroups & 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.

Go deeper:Subgroup Highlighter

Select a subgroup and watch it light up on the Cayley diagram. Notice how subgroup elements cluster together.

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.

Coset Coloring

Choose a group and a subgroup to color each of its cosets. Every element gets exactly one color, and every color covers the same number of elements.

Coset 1 (1 elements)
Coset 2 (1 elements)
Coset 3 (1 elements)
Coset 4 (1 elements)
Coset 5 (1 elements)
Coset 6 (1 elements)

All cosets have equal size (1). Normal subgroup.

Go deeper:Coset Coloring

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.

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.

Lagrange's Theorem

Choose a group and compare each subgroup's order with the group's order. Every subgroup order divides it.

Lagrange's Theorem: The order of every subgroup divides the order of the group.

Subgroup|H||G||G|/|H|Divides?
H0166Yes
H1263Yes
H2263Yes
H3263Yes
H4362Yes
H5661Yes

Subgroup orders as fractions of |G| = 6:

1
2
2
2
3
6
Go deeper:Lagrange's Theorem

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.

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.

Key Takeaways

  • Subgroups -- smaller groups living inside larger ones
  • Cosets -- translates of a subgroup that partition the group into equal-size pieces
  • Lagrange's theorem -- |H| always divides |G|
  • Index -- [G:H] = |G|/|H| = number of cosets

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