Lagrange's Theorem

Intro to Proofs · Axiom Academy

Why a subgroup's size must divide the group's — proven by cutting the group into equal-sized cosets. 1. The Claim: Size Divides Size Let H be a subgroup of a finite group G . The theorem packs into one equation, where [G:H] — the index — counts how many distinct cosets of H fit inside G . order of the group = order of the subgroup × number of cosets |G| — the number of elements in G |H| — the number of elements in the subgroup H [G:H] — the index: how many distinct (left) cosets of H there are 2. A Coset Is Just H , Shifted Pick any element . Its left coset gH is the set you get by multiplying every element of H on the left by g . So a coset is a relocated copy of H . Example — if , then , and a different b gives . The map is a bijection from H to gH , so |gH| = |H| for every g . If gh_1 = gh_2 , cancel g to get h_1 = h_2 — distinct elements of H land on distinct elements of gH . Every element of gH has the form gh by definition, so nothing in gH is missed. Taking g = e gives eH = H . The subgroup is the coset of the identity. In the symmetric group S_3 (order 6 ), take (order 2 ). The coset — still exactly 2 elements, just a shifted pair. 3. Cosets Tile the Group — No Overlap, No Gaps Here is the structural heart of the proof. The cosets of H behave like tiles: any two are either identical or completely disjoint , and together they cover all of G . That is exactly what it means to partition G . Cosets are equal or disjoint: either gH = kH or .

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