subgroup

A subgroup is a group living inside a larger group, using the same operation. Think of a smaller wheel turning within a bigger one: it is self-contained and complete on its own, yet it is part of the surrounding structure. The even integers sit inside all the integers this way — add two evens and you still get an even.

Formally, a subset H of a group G is a subgroup if H is itself a group under G's operation. In practice you check three things: H contains the identity, H is closed under the operation (combining two elements of H stays in H), and H is closed under inverses (the inverse of anything in H is also in H). Every group has at least two subgroups: the whole group, and the trivial one containing only the identity.

Subgroups organize a group into understandable pieces. They are the source of cosets, quotient groups, and Lagrange's theorem, which says a finite subgroup's size must divide the size of the whole group. A subset that is closed under the operation but missing inverses or the identity is not a subgroup — all three conditions are needed.