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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.
We can easily distinguish three kinds of permutations of the three blocks, the conjugacy classes of the group: no change (), a group element of order 1; interchanging two blocks: (RG), (RB), (GB), three group elements of order 2; a cyclic permutation of all three blocks: (RGB), (RBG), two group elements of order 3
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 ...
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).
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 ...
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 ...
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. Here the group B is a Borel subgroup and N is the normalizer of a maximal torus contained in B .
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.