The document discusses permutations, detailing how a permutation is a bijective mapping of a finite set, and introduces the permutation group, also known as the symmetric group, which contains all bijective functions from the set. It explains the concepts of transpositions and the distinction between odd and even permutations, leading to the formation of the alternating group, which consists of even permutations. For the specific example of the set {1, 2, 3}, it identifies the group of even permutations as A3, highlighting its structure and order.

1. ALTERNATING
GROUP
Prem Vishwanath Joshi

2. Permutation
Let X be a non empty finite set. The bijective
mapping 𝑓: 𝑋 → 𝑋 is called a permutation.
The number of elements in the finite set X is
known as degree .
Let X= {𝑎1, 𝑎2, 𝑎3 … … … , 𝑎 𝑛}
Then, permutation
𝑓
=
𝑎1 𝑎2
𝑓(𝑎1) 𝑓(𝑎2)
𝑎3 𝑎4
𝑓(𝑎3) 𝑓(𝑎4) … … … .
𝑎 𝑛−1 𝑎 𝑛
𝑓(𝑎 𝑛−1) 𝑓(𝑎 𝑛)

3. Let X = {1, 2, 3} then all possible permutations are;
𝜎𝑜 =
1 2 3
1 2 3
= (1)
𝜎1 =
1 2 3
2 3 1
= (123)
𝜎2 =
1 2 3
3 1 2
= (132)
𝜎3 =
1 2 3
1 3 2
= (23)
𝜎4 =
1 2 3
3 2 1
= (13)
𝜎5 =
1 2 3
2 1 3
= (12)

4. The set of all permutations
𝑆3 = { 1 , 12 , 13 , 23 , 123 ,
(132)} is called the permutation group of
the set X={1,2,3}.
This permutation group has the degree 3
and order (3!=6)
The permutation group is also called the
symmetric group.

5. Permutation Group
Let 𝑋 = 1, 2, 3, … . . , 𝑛 the set of all bijective
functions from X to X is denoted by 𝑆 𝑛 and there
are 𝑛 distinct permutations (elements). This set
𝑆 𝑛 forms a group under the composition of
functions.
This group is called permutation group or
symmetric group of degree n and the order of
symmetric group 𝑆 𝑛 is n!.
So we can write, 𝑂 𝑆 𝑛 = 𝑛!

6. Transpositions
• A cycle of length 2 is called a transposition.
• Every permutation can be written as a product of
transpositions.
1 3 5 7 = 1 7 1 5 1 3 = 𝑜𝑑𝑑 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛
1 3 5 7 9 = 1 9 1 7 1 5 1 3 = 𝑒𝑣𝑒𝑛 𝑝𝑒𝑟.
2 4 6 3 5 6 = 2 6 2 4 3 6 3 5 = 𝑒𝑣𝑒𝑛
That is ;
𝑛 − 𝑐𝑦𝑐𝑙𝑒 → 𝑛 − 1 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛
Note – Even cycle implies odd permutation
Odd cycle implies even permutation.

7. Alternating group or even permutation
group
An alternating group is the group of even
permutations of a finite set.
Let us consider the set X={1, 2, 3}
The permutation group is for this set is;
𝑆3 = { 1 , 12 , 13 , 23 , 123 , (132)}
The set of even permutations in 𝑆 𝑛 is
denoted by 𝐴 𝑛.

8. For the above example;
1 = 1 2 2 1 = 𝑒𝑣𝑒𝑛
1 2 = 𝑂𝑑𝑑
1 3 = 𝑂𝑑𝑑
2 3 = 𝑂𝑑𝑑
1 2 3 = 1 3 1 2 = 𝑒𝑣𝑒𝑛
1 3 2 = 1 2 1 3 = 𝑒𝑣𝑒𝑛
Hence, Alternating Group
𝐴 𝑛 = { 1 , 1 2 3 , 1 3 2 }
Order of alternating group is
𝑛!
2

9. Thank
You