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Euclidean group - Subgroups

Euclidean group - Subgroups: Encyclopedia II - Euclidean group - Subgroups

Types of subgroups of E(n): Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: Oh and Ih. The groups Ih are even maximal among the groups including the next category. Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically ...

Euclidean group: Encyclopedia II - Euclidean group - Subgroups

Euclidean group - Subgroups

Types of subgroups of E(n):

Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: O_h and I_h. The groups I_h are even maximal among the groups including the next category.
Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically discrete. E.g. for 1 ≤ m ≤ n a group generated by m translations in independent directions, and possibly a finite point group. This includes lattices. Examples more general than those are the discrete space groups.
Countably infinite groups with arbitrarily small translations, rotations, or combinations. In this case there are points for which the set of images under the isometries is not closed. Examples of such groups are, in 1D, the group generated by a translation of 1 and one of √2, and, in 2D, the group generated by a rotation about the origin by 1 radian.
Non-countable groups, where there are points for which the set of images under the isometries is not closed. E.g. in 2D all translations in one direction, and all translations by rational distances in another direction.
Non-countable groups, where for all points the set of images under the isometries is closed. E.g.
- all direct isometries that keep the origin fixed, or more generally, some point (in 3D called the rotation group)
- all isometries that keep the origin fixed, or more generally, some point (the orthogonal group)
- all direct isometries E⁺(n)
- the whole Euclidean group E(n)
- one of these groups in an m-dimensional subspace combined with a discrete group of isometries in the orthogonal n-m-dimensional space
- one of these groups in an m-dimensional subspace combined with another one in the orthogonal n-m-dimensional space

Examples in 3D of combinations:

all rotations about one fixed axis
ditto combined with reflection in planes through the axis and/or a plane perpendicular to the axis
ditto combined with discrete translation along the axis or with all isometries along the axis
a discrete point group, frieze group, or wallpaper group in a plane, combined with any symmetry group in the perpendicular direction
all isometries which are a combination of a rotation about some axis and a proportional translation along the axis; in general this is combined with k-fold rotational isometries about the same axis (k ≥ 1); the set of images of a point under the isometries is a k-fold helix; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes.
for any point group: the group of all isometries which are a combination of an isometry in the point group and a translation; for example, in the case of the group generated by inversion in the origin: the group of all translations and inversion in all points; this is the generalized dihedral group of R³, Dih(R³).