Semidirect Product - Examples

Examples

The dihedral group D_2n with 2n elements is isomorphic to a semidirect product of the cyclic groups C_n and C₂. Here, the non-identity element of C₂ acts on C_n by inverting elements; this is an automorphism since C_n is abelian. The presentation for this group is:

More generally, a semidirect product of any two cyclic groups with generator and with generator is given by a single relation with and coprime, i.e. the presentation:

If and are coprime, is a generator of and, hence the presentation:

gives a group isomorphic to the previous one.

The fundamental group of the Klein bottle can be presented in the form

and is therefore a semidirect product of the group of integers, with itself.

The Euclidean group of all rigid motions (isometries) of the plane (maps f : R2 → R2 such that the Euclidean distance between x and y equals the distance between f(x) and f(y) for all x and y in R2) is isomorphic to a semidirect product of the abelian group R2 (which describes translations) and the group O(2) of orthogonal 2×2 matrices (which describes rotations and reflections which keep the origin fixed). n is a translation, h a rotation or reflection. Applying a translation and then a rotation or reflection corresponds to applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying the conjugate of the original translation). Every orthogonal matrix acts as an automorphism on R2 by matrix multiplication.

The orthogonal group O(n) of all orthogonal real n×n matrices (intuitively the set of all rotations and reflections of n-dimensional space which keep the origin fixed) is isomorphic to a semidirect product of the group SO(n) (consisting of all orthogonal matrices with determinant 1, intuitively the rotations of n-dimensional space) and C₂. If we represent C₂ as the multiplicative group of matrices {I, R}, where R is a reflection of n dimensional space which keeps the origin fixed (i.e. an orthogonal matrix with determinant –1 representing an involution), then φ : C₂ → Aut(SO(n)) is given by φ(H)(N) = H N H–1 for all H in C₂ and N in SO(n). In the non-trivial case ( H is not the identity) this means that φ(H) is conjugation of operations by the reflection (a rotation axis and the direction of rotation are replaced by their "mirror image").

The group of semilinear transformations on a vector space V over a field, often denoted, is isomorphic to a semidirect product of the linear group (a normal subgroup of ), and the automorphism group of . The latter can be embedded as a subgroup of by considering the set .