Quotient Group - Properties

Properties

The quotient group G / G is isomorphic to the trivial group (the group with one element), and G / {e} is isomorphic to G.

The order of G / N, by definition the number of elements, is equal to |G : N|, the index of N in G. If G is finite, the index is also equal to the order of G divided by the order of N. Note that G / N may be finite, although both G and N are infinite (e.g. Z / 2Z).

There is a "natural" surjective group homomorphism π : G → G / N, sending each element g of G to the coset of N to which g belongs, that is: π(g) = gN. The mapping π is sometimes called the canonical projection of G onto G / N. Its kernel is N.

There is a bijective correspondence between the subgroups of G that contain N and the subgroups of G / N; if H is a subgroup of G containing N, then the corresponding subgroup of G / N is π(H). This correspondence holds for normal subgroups of G and G / N as well, and is formalized in the lattice theorem.

Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.

If G is abelian, nilpotent or solvable, then so is G / N.

If G is cyclic or finitely generated, then so is G / N.

If N is contained in the center of G, then G is called the central extension of the quotient group.

If H is a subgroup in a finite group G, and the order of H is one half of the order of G, then H is guaranteed to be a normal subgroup, so G / H exists and is isomorphic to C₂. This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups.

Every finitely generated group is isomorphic to a quotient of a free group.

Sometimes, but not necessarily, a group G can be reconstructed from G / N and N, as a direct product or semidirect product. The problem of determining when this is the case is known as the extension problem. An example where it is not possible is as follows. Z₄ / { 0, 2 } is isomorphic to Z₂, and { 0, 2 } also, but the only semidirect product is the direct product, because Z₂ has only the trivial automorphism. Therefore Z₄, which is different from Z₂ × Z₂, cannot be reconstructed.