symmetric group on three letters


This example is of the symmetric groupMathworldPlanetmathPlanetmath on 3 letters, usually denoted by S3. Here, we are considering the set of bijective functions on the set A={1,2,3} which naturally arise as the set of permutationsMathworldPlanetmath on A. Our binary operationMathworldPlanetmath is function composition which results in a new bijective function. This example develops the table for S3. We start by listing the elements of our group. These elements are listed according to the second method as described in the entry on permutation notation.

e=(1 2 31 2 3)       r=(1 2 32 1 3)
a=(1 2 32 3 1)       s=(1 2 33 2 1)
b=(1 2 33 1 2)       t=(1 2 31 3 2)

Here, our group is just S3={e,a,b,r,s,t}. Now we can start to multiply and then fill in the table. First, we calculate the square of each element.

a2=(1 2 32 3 1)⁢(1 2 32 3 1)=(1 2 33 1 2)=b
b2=(1 2 33 1 2)⁢(1 2 33 1 2)=(1 2 32 3 1)=a
r2=(1 2 32 1 3)⁢(1 2 32 1 3)=(1 2 31 2 3)=e
s2=(1 2 33 2 1)⁢(1 2 33 2 1)=(1 2 31 2 3)=e
t2=(1 2 31 3 2)⁢(1 2 31 3 2)=(1 2 31 2 3)=e

Next, we will fill in the upper right 3⁢x⁡3 block, we only need a⁢b and b⁢a since we can use the fact that there can be no repetition in any row or column.

a⁢b=(1 2 32 3 1)⁢(1 2 33 1 2)=(1 2 31 2 3)=e
b⁢a=(1 2 33 1 2)⁢(1 2 32 3 1)=(1 2 31 2 3)=e

The other 3⁢x⁡3 blocks are also similar. Now continuing with the upper left 3 x 3 block, we go through the table again using the fact that there can be no repetition in any row or column.

a⁢r=(1 2 32 3 1)⁢(1 2 32 1 3)=(1 2 33 2 1)=s
a⁢s=(1 2 32 3 1)⁢(1 2 33 2 1)=(1 2 31 3 2)=t

Similarly, we completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the final blocks of the table.

r⁢a=(1 2 32 1 3)⁢(1 2 32 3 1)=(1 2 31 3 2)=t
r⁢b=(1 2 32 1 3)⁢(1 2 33 1 2)=(1 2 33 2 1)=s
s⁢a=(1 2 33 2 1)⁢(1 2 32 3 1)=(1 2 32 1 3)=r
s⁢r=(1 2 33 2 1)⁢(1 2 32 1 3)=(1 2 32 3 1)=a

Finally, we fill in the table using the calculated values above.

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Title symmetric group on three letters
Canonical name SymmetricGroupOnThreeLetters
Date of creation 2013-03-22 15:52:24
Last modified on 2013-03-22 15:52:24
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Example
Classification msc 20B30