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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 ...
An outer automorphism interchanges the two classes of M 22 groups. This outer automorphism is realized on McL embedded as a subgroup of Co 3 . A convenient representation of M 22 is in permutation matrices on the last 22 coordinates; it fixes a 2-2-3 triangle with vertices the origin and the type 2 points x = (−3, 1 23 ) and y = (−4,-4,0 22
The standard quotient theorem of set theory then gives a natural bijection between G / G x and Gx. Specifically, the bijection is given by hG x ↦ h · x. This result is known as the orbit-stabilizer theorem. In the two cases of a small orbit, the stabilizer is non-trivial.
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 ...
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
Suzuki showed that the Suzuki group has q+3 conjugacy classes. Of these, q+1 are strongly real, and the other two are classes of elements of order 4. q 2 +1 Sylow 2-subgroups of order q 2, of index q–1 in their normalizers. 1 class of elements of order 2, 2 classes of elements of order 4.
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 ...