Subgroups and Cosets: Definitions, Lagrange's Theorem,

A subgroup H of a group G is a subset that is itself a group under G's operation — for instance, the even integers (2ℤ, +) inside (ℤ, +). Cosets are the "shifted copies" of H: for each g ∈ G, the left coset gH = {g ∗ h : h ∈ H}. The cosets partition G into equal-sized blocks, like slicing a cake into identical pieces.

This slicing is the key to Lagrange's theorem: for finite G, the order of any subgroup divides the order of G. That single divisibility fact restricts which subgroups can exist, proves Fermat's little theorem, and is the first step toward quotient groups.

Key Concepts

• Subgroup: H ⊆ G is a subgroup if it is non-empty and closed under the operation and inverses (for finite H, closure alone suffices).
• Subgroup test: H ≠ ∅, and a, b ∈ H ⟹ ab⁻¹ ∈ H — one condition doing the work of three.
• Left coset: gH = {gh : h ∈ H}; right coset Hg = {hg : h ∈ H}.
• Partition property: distinct cosets are disjoint, and every element of G lies in exactly one coset.
• Index: [G : H], the number of distinct cosets of H in G.
• Lagrange's theorem: for finite G, |G| = |H| · [G : H]; hence |H| divides |G|.

Formulas

• Subgroup test: H ≤ G ⟺ H ≠ ∅ and ∀a, b ∈ H: ab⁻¹ ∈ H.
• Lagrange: |G| = |H| × [G : H].
• Coset equality: aH = bH ⟺ b⁻¹a ∈ H (equivalently a⁻¹b ∈ H for right cosets).
• Element order divides group order: for finite G, a^|G| = e for every a ∈ G.

Worked Example

Let G = ℤ₁₂ (integers mod 12 under addition) and H = {0, 4, 8}. Verify H is a subgroup, list its cosets, and confirm Lagrange's theorem.

1. Subgroup test: 0 ∈ H; closure: 4+4 = 8, 4+8 = 12 ≡ 0, 8+8 = 16 ≡ 4 (mod 12) ✓; inverses: −4 ≡ 8 ∈ H, −8 ≡ 4 ∈ H ✓. So H ≤ G with |H| = 3.
2. Cosets: 0+H = {0, 4, 8}; 1+H = {1, 5, 9}; 2+H = {2, 6, 10}; 3+H = {3, 7, 11}. Cosets 4+H onward repeat these four.
3. The four cosets are disjoint and cover all 12 elements — a partition ✓.
4. Lagrange check: |G| = 12, |H| = 3, [G:H] = 4, and 12 = 3 × 4 ✓.
5. Bonus: the element 5 has order 12/gcd(5,12) = 12, and 12 divides |G| = 12 ✓.

Common Misconceptions

• "Any subset containing the identity is a subgroup." No — {0, 1} inside (ℤ₄, +) contains 0 but 1+1 = 2 ∉ {0,1}, so closure fails.
• "Cosets are subgroups." Only the coset eH = H itself is a subgroup; the others do not even contain the identity.
• "Lagrange's theorem works backwards." It does not: A₄ has order 12 but no subgroup of order 6, so a divisor of |G| need not occur as a subgroup order.

Real-World Applications

• Number theory: Fermat's little theorem (a^{p−1} ≡ 1 mod p) follows from Lagrange applied to (ℤ/pℤ)*.
• Coding theory: cosets of linear codes are the syndromes used in error correction.
• Quotient groups: cosets become the elements of G/H once H is normal — the next topic.

Subgroups are groups-within-groups, cosets are their shifted copies partitioning G, and Lagrange's theorem ties the sizes together: |G| = |H| · [G:H]. The ℤ₁₂ example shows the full machinery — subgroup test, four cosets, divisibility — in one compact computation.