Subgroup and Coset Calculator

A subgroup of G is a subset H that is a group under the operation of G. Three things have to be checked: the identity is in H; the product of two elements of H is in H; the inverse of an element of H is in H. Associativity needs no check: (ab)c = a(bc) holds for all elements of G, so in particular for those of H. The tester checks the subset exactly as you chose it and adds nothing to it.

⟨a₁, …, aₖ⟩ is the smallest subgroup containing the chosen elements: everything that can be written as a product of them. That is a different question from testing a subset: the generators need not form a subgroup themselves. In a finite group the inverse of a is a power of a, so products of the generators already give everything. The empty set generates {e}.

For a subgroup H and an element g, the left coset is gH = {gh : h ∈ H} and the right coset is Hg = {hg : h ∈ H}. Two elements give the same left coset exactly when g⁻¹k ∈ H, and the same right coset exactly when gk⁻¹ ∈ H. In an abelian group the two kinds agree; in general they need not.

Every element g lies in its own coset gH, two cosets are either equal or disjoint, and h ↦ gh matches H with gH one to one. So the left cosets cut G into pieces of the same size |H|, and counting gives Lagrange’s theorem: |G| = |H| [G : H], where the index [G : H] is the number of cosets. The same holds for right cosets, and there are as many of them.

The conjugate of h by g is ghg⁻¹. A subgroup is normal, H ⊴ G, when conjugation never leads out of it: gHg⁻¹ = H for every g. That is the same as gH = Hg for every g: the left and right partitions coincide. Subgroups of abelian groups, subgroups of index 2, and the center are always normal.

For a normal subgroup the cosets themselves form a group G/H under (gH)(kH) = (gk)H, of order [G : H]. The rule uses representatives, so it must give the same coset whichever representatives are taken. If H is normal it does: (gh)(kh′) = gk (k⁻¹hk) h′, and k⁻¹hk ∈ H. If H is not normal it does not, and the page shows two choices that disagree: the cosets are still there, but they do not multiply.

The intersection of two subgroups is a subgroup. Their union usually is not; the smallest subgroup containing both is the join ⟨H ∪ K⟩. The set of products HK has |H||K| / |H ∩ K| elements and is a subgroup exactly when HK = KH, for instance when one of the two is normal; then it is the join. Ordered by inclusion, the subgroups form a lattice. Its picture, the Hasse diagram, has a line only where one subgroup lies directly inside another.