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A left coset of subgroup H in group G is a set of the form aH = {ah : h ∈ H}. Cosets partition G into equal-sized disjoint subsets. Lagrange's theorem states that the order of H divides the order of G, and [G : H] = |G| / |H| is the index of H in G.
Think of a coset as a "shifted copy" of the subgroup H. You already know from your study of subgroups that H is a subset of G closed under the group operation, and from group isomorphisms that structure can be preserved under mappings. A coset aH takes every element h of H and applies a fixed group element a to it on the left: aH = {ah : h ∈ H}. The result is not usually a subgroup itself — it is a translate of H, the same shape but sitting in a different part of G.
The crucial fact about cosets is that they partition the group: every element of G belongs to exactly one left coset of H. This follows from two observations. First, every a belongs to its own coset aH (since the identity e is in H, so ae = a ∈ aH). Second, any two cosets are either identical or completely disjoint — there is no partial overlap. You can verify this yourself: if x belongs to both aH and bH, then you can write x = ah₁ = bh₂, which means a and b are in the same coset. The cosets tile G perfectly, like congruence classes mod n tile the integers.
Lagrange's Theorem is the immediate payoff: since all cosets have the same size as H, and since they partition G without overlap, the number of cosets times |H| must equal |G|. In symbols, |G| = [G : H] · |H|, which means |H| divides |G|. This is a powerful divisibility constraint on subgroup orders. For example, a group of order 15 cannot have a subgroup of order 4 or 6 — only orders 1, 3, 5, and 15 are even candidates. This filters the possibilities before you do any detailed analysis.
The index [G : H] counts how many distinct cosets H has in G. For finite groups it equals |G|/|H|. For infinite groups (like the integers Z under the subgroup nZ), the index still makes sense — it counts the equivalence classes, which recover the familiar congruence classes mod n. Lagrange's theorem is the reason why the order of any group element divides the order of the group: the cyclic subgroup generated by an element has order equal to the element's order, and that subgroup's order must divide |G|. This connects cosets to the structure theory you will use throughout the rest of abstract algebra.