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Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class with elements of order 6", and "6B" would be a different conjugacy class with elements of order 6; the conjugacy class 1A is the conjugacy class of the identity which has order 1.
The manipulations of the Rubik's Cube form the Rubik's Cube group. The Rubik's Cube group (,) represents the structure of the Rubik's Cube mechanical puzzle.Each element of the set corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces.
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 1D, all reflections are in the same class. In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class. In 3D:
All the reflections are conjugate to each other whenever n is odd, but they fall into two conjugacy classes if n is even. If we think of the isometries of a regular n-gon: for odd n there are rotations in the group between every pair of mirrors, while for even n only half of the mirrors can be reached from one by these rotations. Geometrically ...
In D 12 reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes. By contrast, if n is even, then 4 divides the order of the group, and the subgroups of order 2 are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or ...
Any element of conjugacy class 3A 56 generates Sylow 3-subgroup. Any element from the conjugacy classes 7A 24, 7B 24 generates the Sylow 7-subgroup. The Sylow 2-subgroup is a dihedral group of order 8. It can be described as centralizer of any element from the conjugacy class 2A 21.
The action of L 2 (11) can be seen algebraically as due to an exceptional inclusion L 2 (5) L 2 (11) – there are two conjugacy classes of subgroups of L 2 (11) that are isomorphic to L 2 (5), each with 11 elements: the action of L 2 (11) by conjugation on these is an action on 11 points, and, further, the two conjugacy classes are related by ...
By definition, an element is central whenever its conjugacy class contains only the element itself; i.e. Cl(g) = {g}. The center is the intersection of all the centralizers of elements of G: = (). As centralizers are subgroups, this again shows that the center is a subgroup.