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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.
Thus the conjugacy class within the Euclidean group E(n) of inversion in a point is the set of inversions in all points. Since a combination of two inversions is a translation, the conjugate closure of a singleton containing inversion in a point is the set of all translations and the inversions in all points.
S 6 has exactly one (class) of outer automorphisms: Out(S 6) = C 2. To see this, observe that there are only two conjugacy classes of S 6 of size 15: the transpositions and those of class 2 3. Each element of Aut(S 6) either preserves each of these conjugacy classes, or exchanges them. Any representative of the outer automorphism constructed ...
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For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups. Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive ) algebraic groups, in Jacques Tits ' theory of groups with a ( B , N ) pair .
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:
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The conjugacy classes of T are: identity; 4 × rotation by 120°, order 3, cw; 4 × rotation by 120°, order 3, ccw; 3 × rotation by 180°, order 2; The octahedral group of order 24, rotational symmetry group of the cube and the regular octahedron. It is isomorphic to S 4. The conjugacy classes of O are: identity; 6 × rotation by ±90 ...