Search results
Results from the WOW.Com Content Network
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 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).
Characters are class functions, that is, they each take a constant value on a given conjugacy class. More precisely, the set of irreducible characters of a given group G into a field F form a basis of the F-vector space of all class functions G → F. Isomorphic representations have the same characters.
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 above exchanges the conjugacy classes, whereas an index 2 subgroup ...
Thus the class of (2,2)-cycles contains 15 elements, while the icosidodecahedron has 30 vertices. The two conjugacy classes of twelve 5-cycles in A 5 are represented by two icosahedra, of radii 2 π /5 and 4 π /5, respectively. The nontrivial outer automorphism in Out(A 5) ≃ Z 2 interchanges these two classes and the corresponding icosahedra.
The dihedral group D 2 is generated by the rotation r of 180 degrees, and the reflection s across the x-axis. The elements of D 2 can then be represented as {e, r, s, rs}, where e is the identity or null transformation and rs is the reflection across the y-axis. The four elements of D 2 (x-axis is vertical here) D 2 is isomorphic to the Klein ...
Note that χ ρ is constant on conjugacy classes, that is, χ ρ (π) = χ ρ (σ −1 πσ) for all permutations σ. Over other fields the situation can become much more complicated. If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KS n is semisimple. In these cases the irreducible ...
The conjugacy classes of S n correspond to the cycle types of permutations; that is, two elements of S n are conjugate in S n if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S 5, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate