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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.
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 ...
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 ...
In particular, all complex representations decompose as a direct sum of irreps, and the number of irreps of is equal to the number of conjugacy classes of . [ 5 ] The irreducible complex representations of Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } are exactly given by the maps 1 ↦ γ {\displaystyle 1\mapsto \gamma } , where γ ...
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.
The set of integer-valued class functions on G, Z([G]), is a commutative ring, finitely generated over . All of its elements are thus integral over Z {\displaystyle \mathbb {Z} } , in particular the mapping u which takes the value 1 on the conjugacy class of g and 0 elsewhere.
The Deligne–Simpson problem is the following realisation problem: For which tuples of conjugacy classes in GL(n, C) do there exist irreducible tuples of matrices M j from these classes satisfying the above relation?
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