Determination of multiplication table of symmetric group:S3
This page describes the process used for the determination of specific information related to a particular group. The information type is multiplication table and the group is symmetric group:S3.
View all pages that describe how to determine multiplication table of particular groups  View all specific information about symmetric group:S3
The purpose of this page is to give a detailed description of the construction of the multiplication table of symmetric group:S3. The survey article is meant as a walkthrough of the entire process rather than a summary and is written for people relatively new to group theory. If you're looking for terse summary descriptions, check out symmetric group:S3 and element structure of symmetric group:S3.
Contents
The final table
We first provide the final multiplication table in cycle decomposition notation and also oneline notation for the group acting on the set . The convention followed here is that the column element is on the right and the row element is on the left, and functions act on the left. Hence, to determine the effect of the composite permutation on any element of , we must first apply the permutation given by the column element and then apply the permutation given by the row element.
Cycle decomposition notation
Element  

Oneline notation
Element  123  213  132  321  231  312 

123  123  213  132  321  231  312 
213  213  123  231  312  132  321 
132  132  312  123  231  321  213 
321  321  231  312  123  213  132 
231  231  321  213  132  312  123 
312  312  132  321  213  123  231 
For this article, we follow the left action convention, which is standard in most introductory courses and treatments, although group theorists often uses right action because of the convenience of exponential notation.
Review of preliminaries
What multiplication of permutations means
Suppose and are (possibly equal) permutations on a set . is also a permutation. For , we define:
In other words, we first apply to , and locate the answer as an element of . Having done this, we apply to that element. The final answer we get is where should send .
Thus, to describe , we need to apply the above procedure to every element .
We will restrict our attention to , and in fact to the case , though our initial remarks apply to other .
What it means with oneline notation
It is pretty easy to multiply two permutations written in oneline notation. The first step is to convert the oneline notation to twoline notation. Recall that the oneline notation for a permutation on the set simply lists the images . The twoline notation is:
The oneline notation is obtained by suppressing the top line of the twoline notation.
When multiplying, the twoline notation is operationally easier. To find the twoline notation for , we have to find for each . We do this by first looking up the entry under in the twoline notation for . Call that . We now look up the entry under in the twoline notation for . Whatever answer we get, we write it as the entry under in the twoline notation for . We do this for each . To retrieve the oneline notation, we simply remove the top line.
What it means with cycle decomposition notation
Further information: cycle decomposition, understanding the cycle decomposition
The cycle decomposition of a permutation breaks it up as a product of disjoint cycles. The permutations and that we are multiplying each has its own cycle decomposition. Composing them could give a permutation that has a very different cycle decomposition from either of them. The exception occurs when both and are equal, or are powers of each other  in this case, the cycle structure of the product looks very similar to that of and . For more, see Understanding the cycle decomposition#Computing the powers of a permutation.
All the 36 multiplications
We proceed column major (i.e., we do all the multiplications in a column and then move to the next column), providing full explanations with the twoline notation, along with any shortcut explanations where they exist, for each of the multiplications.
First column and first row
The first column of the multiplication table considers multiplication where the second element being multiplied is the identity element (the permutation that sends 1 to 1, 2 to 2, and 3 to 3). By the definition of identity element, the product is just the first element. Therefore, the first column coincides with the row headers column
Similarly, the first row of the multiplication table considers multiplication where the first element is the identity . Thus, the first row coincides with the column headers row.
The partial multiplication table constructed thus far is:
Element  

?  ?  ?  ?  ?  
?  ?  ?  ?  ?  
?  ?  ?  ?  ?  
?  ?  ?  ?  ?  
?  ?  ?  ?  ? 
Second column: multiplication with
We need to consider multiplications where the second group element being multiplied is the permutation . In oneline notation, this is the element 213, and in twoline notation, this is the element:
We consider the multiplication of this with different elements:
Element multiplied on the left (in cycle decomposition, oneline, and twoline notation)  Tracing what happens under the multiplication with  Product 










The multiplication table so far is:
Element  

?  ?  ?  ?  
?  ?  ?  ?  
?  ?  ?  ?  
?  ?  ?  ?  
?  ?  ?  ? 
Sanity checks:
 No two elements in the same column are equal. In particular, each completed column lists each group element exactly once.
 No two elements in the same row are equal.
Third column: multiplication with
We need to consider multiplications where the second group element being multiplied is the permutation . In oneline notation, this is the element 132, and in twoline notation, this is the element:
We consider the multiplication of this with different elements:
Element multiplied on the left (in cycle decomposition, oneline, and twoline notation)  Tracing what happens under the multiplication with  Product 










The multiplication table so far is:
Element  

?  ?  ?  
?  ?  ?  
?  ?  ?  
?  ?  ?  
?  ?  ? 
Sanity checks:
 No two elements in the same column are equal. In particular, each completed column lists each group element exactly once.
 No two elements in the same row are equal.
Fourth column: multiplication with
We need to consider multiplications where the second group element being multiplied is the permutation . In oneline notation, this is the element 321, and in twoline notation, this is the element:
We consider the multiplication of this with different elements:
Element multiplied on the left (in cycle decomposition, oneline, and twoline notation)  Tracing what happens under the multiplication with  Product 










The multiplication table so far is:
Element  

?  ?  
?  ?  
?  ?  
?  ?  
?  ? 
Sanity checks:
 No two elements in the same column are equal. In particular, each completed column lists each group element exactly once.
 No two elements in the same row are equal.
Fifth column: multiplication with
We need to consider multiplications where the second group element being multiplied is the permutation . In oneline notation, this is the element 231, and in twoline notation, this is the element:
We consider the multiplication of this with different elements:
Element multiplied on the left (in cycle decomposition, oneline, and twoline notation)  Tracing what happens under the multiplication with  Product 










The multiplication table so far is:
Element  

Sanity checks:
 No two elements in the same column are equal. In particular, each completed column lists each group element exactly once.
 No two elements in the same row are equal.
Sixth column: multiplication with
We need to consider multiplications where the second group element being multiplied is the permutation . In oneline notation, this is the element 312, and in twoline notation, this is the element:
We consider the multiplication of this with different elements:
Element multiplied on the left (in cycle decomposition, oneline, and twoline notation)  Tracing what happens under the multiplication with  Product 










The multiplication table is now complete:
Element  

Postcompletion sanity checks
Latin square
 Every column has each group element exactly once.
 Every row has each group element exactly once.
Inverses
 Each of the transpositions (, , and ) is equal to its own inverse. This makes natural sense.
 The two 3cycles are inverses as well as squares of each other.
Alternating group as subgroup
The identity element and the two 3cycles form the subgroup A3 in S3, i.e., the alternating group of degree three, which is isomorphic to cyclic group:Z3 (these are the even permutations). The nonidentity coset of this subgroup comprises the three 2transpositions (these are the odd permutations). It's easy to eyeball and see that:
 The product of any two even permutations is even.
 The product of any two odd permutations is even.
 The product of an even permutation and an odd permutation is odd.
 The product of an odd permutation and an even permutation is odd.
This is easiest to see if we reorder the rows and columns so that the even permutations appear first and the odd permutations appear later:
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