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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.
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
The conjugacy classes of the full octahedral group, O h ≅ S 4 × C 2, are: inversion; 6 × rotoreflection by 90° 8 × rotoreflection by 60° 3 × reflection in a plane perpendicular to a 4-fold axis; 6 × reflection in a plane perpendicular to a 2-fold axis; The conjugacy classes of full icosahedral symmetry, I h ≅ A 5 × C 2, include also ...
The entries in the same row are in the same conjugacy class. Every entry appears once in each column, as seen in the file below. Every entry appears once in each column, as seen in the file below. The positions of permutations with inversion sets symmetric to each other have positions in the table that are symmetric to each other.
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the general linear group GL n (n x n invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup.
Conjugacy class in group theory, related to matrix similarity in linear algebra; Conjugation (group theory), the image of an element under the conjugation homomorphisms; Conjugate closure, the image of a subgroup under the conjugation homomorphisms; Conjugate words in combinatorics; this operation on strings resembles conjugation in groups
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