Subgroup

A subgroup is a subset of a group that is itself a group under the same operation. Subgroups are the first lens through which we study the internal structure of a group.

Definition

A non-empty subset H ⊆ G of a group G is a subgroup, written H ≤ G, if

• a, b ∈ H ⟹ ab ∈ H (closed under the operation), and
• a ∈ H ⟹ a⁻¹ ∈ H (closed under inverses).

Equivalently (finite case): a non-empty subset closed under the operation. The identity lies in every subgroup.

H is a proper subgroup if H ≠ G; trivial if H = {e}.

For a subset S ⊆ G, the subgroup generated by S, written ⟨S⟩, is the smallest subgroup containing S — equivalently, the intersection of all subgroups containing S, or the set of all finite products of elements of S and their inverses. S generates G if ⟨S⟩ = G.

Examples

• Cyclic subgroup ⟨a⟩ = {e, a, a², …} of order |a|.
• Alternating group Aₙ ≤ Sₙ — the even permutations. Index 2.
• Rotations in Dₙ form a cyclic subgroup Cₙ ≤ Dₙ of index 2.
• Center Z(G) = {z ∈ G : zg = gz for all g} is always a subgroup. See center.
• Diagonal {(g, g) : g ∈ G} ≤ G × G.

Properties

• {e} and the whole group G are always subgroups of G.
• An intersection of subgroups is a subgroup — for arbitrarily many, even infinitely many.
• A union of subgroups is usually not a subgroup: ⟨a⟩ ∪ ⟨b⟩ is a subgroup only when one of ⟨a⟩, ⟨b⟩ contains the other.
• Lagrange's theorem: in a finite group, |H| divides |G| for every subgroup H ≤ G.
• The subgroups of a group form a lattice under inclusion — meet is intersection, join is the subgroup generated by the union — and a finite group has only finitely many.
• Every subgroup and every quotient of an abelian group is again abelian.
• A normal subgroup times any subgroup is a subgroup: if N ⊴ G and M ≤ G, then NM = {nm : n ∈ N, m ∈ M} is a subgroup; if both are normal, so is NM.

See also

• Order — |⟨a⟩| = |a|.
• Normal subgroup — subgroups that are stable under conjugation.
• Center — the canonical abelian subgroup.