3-cycle in symmetric group:S3

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Template:Particular automorphism class

We consider the group G defined as symmetric group:S3, i.e., the symmetric group of degree three, which for convenience we take to be the symmetric group acting on the set \{ 1, 2, 3 \}.

We are interested in the conjugacy class of 3-cycles in this group, i.e., permutations that move all elements and have order three. The 3-cycles form a single conjugacy class and also form a single orbit under the action of the automorphism group of S_3.

The full list of elements in the conjugacy class is:

\{ (1,2,3), (1,3,2) \}

Arithmetic functions

Function Value Explanation
order of the whole group 6
size of conjugacy class 2
size of automorphism class 2
number of conjugacy classes in automorphism class 1
order of elements in conjugacy class 3

Description in alternative interpretations of the whole group

Interpretation of G Description of conjugacy class Verification of size computation Verification of element order computation More information
As the symmetric group of degree three corresponds to the partition 3 \frac{3!}{3} = 2 \operatorname{lcm}\{ 3 \} = 3 element structure of symmetric groups, element structure of symmetric group:S3#Interpretation as symmetric group, conjugacy class size formula in symmetric group
As the dihedral group D_{2n}, n = 3, 2n = 6 (n odd) the generator of the cyclic piece of order n 2 (valid for n \ge 3) n = 3 element structure of dihedral groups, element structure of symmetric group:S3#Interpretation as dihedral group
As the special linear group SL(2,q), q = 2 (q even) the unique conjugacy class of semisimple elements not diagonalizable over the field itself q(q - 1) = 2(2 - 1) = 2 Must divide q^2 - 1 in general, in this case, it is q^2 - 1 = 2^2 - 1 = 3 element structure of special linear group of degree two over a finite field, element structure of symmetric group:S3#Interpretation as general linear group of degree two
As the general affine group GA(1,q), q = 3 the conjugacy class of all maps of the form x \mapsto x + b, b \in \mathbb{F}_3^\ast. q - 1 = 2 q = 3 element structure of general affine group of degree one over a finite field, element structure of symmetric group:S3#Interpretation as general affine group of degree one