Algebra 10 | Subgroups

A subgroup is a smaller group hiding inside a bigger group: pick a nonempty subset H of a group G, keep the same binary operation, and require that H itself becomes a group. The payoff is practical—once the subgroup conditions are met, there’s no need to re-check every group axiom from scratch, because associativity and the operation structure already come from G.

The video starts with the intuition of “embedding” one algebraic structure into another. A concrete example anchors the idea: the real numbers R under addition contain the integers Z under addition, and Z forms a subgroup of R. This sets up the central question for the lesson: how to recognize when a subset of a group is actually a subgroup, and what properties must hold.

Formally, a group G consists of a set with an associative binary operation, an identity element e, and inverses for every element. A subset H ⊆ G (with H ≠ ∅) is called a subgroup when H, using the same binary operation restricted to elements of H, is itself a group. The key simplification is that associativity automatically holds on H because it already holds on G. That leaves only two essential checks: (1) closure under the operation and (2) closure under inverses.

That leads to the main criterion: H is a subgroup of G if and only if for all A, B in H, the product A ∘ B lies in H, and for every A in H, the inverse A⁻¹ also lies in H. The lesson justifies why these two conditions are enough. If H already forms a group, closure is immediate, and inverses stay inside H. Conversely, if closure under the operation and inverses holds, then associativity is inherited from G, and the identity element e must land in H: pick any a in H (possible because H is nonempty), then a⁻¹ is in H by the inverse-closure rule, and combining a with a⁻¹ stays in H by closure—yielding e. With identity and inverses in place, H becomes a group, hence a subgroup.

The video then frames subgroup examples in two extremes. Every group G has at least two “trivial” subgroups: {e} (the smallest possible subgroup) and G itself (the largest). For nontrivial groups, there are at least these two distinct subgroups.

Finally, it gives an infinite family of concrete subgroups in the integers. For any natural number m, the set mZ of multiples of m—written as {m·k : k ∈ Z}—forms a subgroup of Z under addition. This produces infinitely many subgroups of Z. The discussion also points ahead to a broader theme: once a subgroup H exists, it becomes possible to form new groups built from cosets, leading toward the quotient construction G/H, which will be treated next.