commutes with any linear transformation).

In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones.

For n twice an odd number, the abstract group is isomorphic with the direct product of and .Generally, if m divides n, then has n/m subgroups of type , and one subgroup _m. Therefore, the total number of subgroups of (n >= 1),

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