Dihedral Group of Order 6 - Subgroups

Subgroups

From Lagrange's theorem we know that any non-trivial subgroup has order 2 or 3. In fact the two cyclic permutations of all three blocks, with the identity, form a subgroup of order 3, index 2, and the swaps of two blocks, each with the identity, form three subgroups of order 2, index 3.

The first-mentioned is {,(RGB),(RBG)}, the alternating group A₃.

The left cosets and the right cosets of A₃ are both that subgroup itself and the three swaps.

The left cosets of {,(RG)} are:

• that subgroup itself
• {(RB),(RGB)}
• {(GB),(RBG)}

The right cosets of {(RG),} are:

• that subgroup itself
• {(RBG),(RB)}
• {(RGB),(GB)}

Thus A₃ is normal, and the other three non-trivial subgroups are not. The quotient group G / A₃ is isomorphic with C₂.

, a semidirect product, where H is a subgroup of two elements: and one of the three swaps.

In terms of permutations the two group elements of G/ A₃ are the set of even permutations and the set of odd permutations.

If the original group is that generated by a 120° rotation of a plane about a point, and reflection with respect to a line through that point, then the quotient group has the two elements which can be described as the subsets "just rotate (or do nothing)" and "take a mirror image".

Note that for the symmetry group of a square, an uneven permutation of vertices does not correspond to taking a mirror image, but to operations not allowed for rectangles, i.e. 90° rotation and applying a diagonal axis of reflection.