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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.
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 ...
First, show that any automorphism that preserves the conjugacy class of transpositions is an inner automorphism. (This also shows that the outer automorphism of S 6 is unique; see below.) Note that an automorphism must send each conjugacy class (characterized by the cyclic structure that its elements share) to a (possibly different) conjugacy ...
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).
For example, if the generating set has elements then each vertex of the Cayley graph has incoming and outgoing directed edges. In the case of a symmetric generating set S {\displaystyle S} with r {\displaystyle r} elements, the Cayley graph is a regular directed graph of degree r . {\displaystyle r.}
By inspection, we can also determine associativity and closure (two of the necessary group axioms); note for example that (ab)a = a(ba) = aba, and (ba)b = b(ab) = bab. The group is non-abelian since, for example, ab ≠ ba. Since it is built up from the basic actions a and b, we say that the set {a, b} generates it. The group has presentation
Thus the conjugacy class within the Euclidean group E(n) of a translation is the set of all translations by the same distance. The smallest subgroup of the Euclidean group containing all translations by a given distance is the set of all translations. So, this is the conjugate closure of a singleton containing a translation.
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 ...