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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 ...
If a finite group G has exactly two conjugacy classes of involutions with representatives t and z, then the Thompson order formula (Aschbacher 2000, 45.6) (Suzuki 1986, 5.1.7) states
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 finite group is abelian if and only if =.; One has = #where () is the number of conjugacy classes of .. If is not abelian then () / (this result is sometimes called the 5/8 theorem [5]) and this upper bound is sharp: there are infinitely many finite groups such that () = /, the smallest one being the dihedral group of order 8.
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The conjugacy classes of T are: identity; 4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321) 4 × rotation by 120° counterclockwise (ditto) 3 × rotation by 180° The rotations by 180°, together with the identity, form a normal subgroup of type Dih 2, with quotient group of type Z 3. The three elements of the ...
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.