Transposition 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 transpositions in this group, i.e., permutations that flip two elements and fix the third element. The transpositions 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), (2,3), (1,3) \}

Arithmetic functions

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

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 2 + 1 \frac{3!}{(2)(1)} = 3 \operatorname{lcm}\{ 2,1 \} = 2 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 conjugacy class of all "reflections" i.e. things outside the cyclic subgroup of index two n = 3 2 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 unipotent elements (note that the conjugacy class is unique when q is even) q^2 - 1 = 2^2 - 1 = 3 characteristic prime, i.e., prime whose power is q. In this case, it is 2 element structure of special linear group of degree two over a finite field, element structure of symmetric group:S3#Interpretation as special 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 2x + b, b \in \mathbb{F}_3. q = 3 order of 2 in \mathbb{F}_3^\ast, which equals 2 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
As the general semilinear group \Gamma L(1,p^2), p = 2, p^2 = 4 the conjugacy class of all maps whose field automorphism part is the non-identity map. Note that for generic p, there are p - 1 such conjugacy classes, but since p = 2, the conjugacy class is unique. p + 1 = 2 + 1 = 3 Since this is the conjugacy class containing the mapping that is simply the field automorphism, it has order 2. element structure of general semilinear group of degree one over a finite field, element structure of symmetric group:S3#Interpretation as general semilinear group of degree one