Normal Subgroup - Examples

Examples

• The subgroup {e} consisting of just the identity element of G and G itself are always normal subgroups of G. The former is called the trivial subgroup, and if these are the only normal subgroups, then G is said to be simple.
• The center of a group is a normal subgroup.
• The commutator subgroup is a normal subgroup.
• More generally, any characteristic subgroup is normal, since conjugation is always an automorphism.
• All subgroups N of an abelian group G are normal, because gN = Ng. A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.
• The translation group in any dimension is a normal subgroup of the Euclidean group; for example in 3D rotating, translating, and rotating back results in only translation; also reflecting, translating, and reflecting again results in only translation (a translation seen in a mirror looks like a translation, with a reflected translation vector). The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.
• In the Rubik's Cube group, the subgroup consisting of operations which only affect the corner pieces is normal, because no conjugate transformation can make such an operation affect an edge piece instead of a corner. By contrast, the subgroup consisting of turns of the top face only is not normal, because a conjugate transformation can move parts of the top face to the bottom and hence not all conjugates of elements of this subgroup are contained in the subgroup.