related to group theory and linear algebra

1. GROUP THEORY AND
LINEAR ALGEBRA
PRESENTED TO: MAMMEHAK
PRESENTED BY: AMENAH GONDAL
CLASS: BS.ED VI0

2. OBJECTIVES:
WE WILL DISCUSS:
TRANSPOSITIONS
EVEN ODD PERMUTATION
PROPERTIES OF PERMUTATIONS
ALTERNATING GROUP

3. TRANSPOSITION:
“A cycle of length 2, that is a cycle of the form (x,y) is called a transposition.”
Thus a permutation of the type
𝑎 𝑏
𝑏 𝑎
Where a and b are interchanged by this permutation and other
elements remain fixed is a transposition.
Also,
(𝑥 𝑦)2
So that,
𝑥 𝑦 = (𝑥 𝑦)−1
= (𝑦 𝑥)

4. EXAMPLE:
Let we have a cycle of length 7
which is (1 2 3 4 5 6 7)
then the transpositions are
(1 2), (13), (1 4), (1 5), (1 6), (1 7).