Today, we will learn about permutation groups, a different way to represent a finite group. Permutation groups can be abelian or non-abelian, though usually finite abelian group are represented in addition form.
Consider the symmetries of the square. We have represented the elements as 2x2 matrices. If we think about the entire xy plane, these matrices rearrange the four quadrants. Usually, the quadrants are given the Roman numerals I, II, III and IV, but I will use the Hindu-Arabic numerals 1, 2, 3 and 4, because in permutation notation, these number are going to be listed without spaces between them.
R0°, the identity I
2 | 1
3 | 4
Nothing moves when the identity is applied, 1 goes to 1, 2 goes to 2, 3 to 3 and 4 to 4. In cycle notation, this could be written
(1)(2)(3)(4)
but if a numeral remains fixed by a permutation, we usually don't write it at all. The standard form of the identity in a permutation group is
(1).
just to show there is something there.
R90°
1 | 4
2 | 3
Here, 1 goes to position 2, 2 to position 3, 3 to position 4 and 4 to position 1. That means we have a single cycle of length 4,
(1234).
R180°
4 | 3
1 and 3 switch places, as do 2 and 4. We have two cycles of length 2,
(13)(24).
2-cycles are often called transpositions. They are the building blocks of all permutations.
R270°
3 | 2
Here, 1 goes to position 4, 4 to position 3, 3 to position 2 and 2 to position 1. That means we have another single cycle of length 4,
(1423)
M0°
3 | 4
1 and 4 switch positions, as do 2 and 3.
(14)(23)
Let me note that it is standard for the first number in a cycle is the smallest value of the set of numbers contain in the cycle.
M45°
4 | 1
3 | 2
This is a single transposition
(24).
We will see that M135° is also a single transposition.
M90°
1 | 2
4 | 3
1 and 2 switch positions, as do 3 and 4, so we get
(14)(23)
M135°
2 | 3
This is a single transposition
(13).
Tomorrow, we will define the operation composition, which is how permutations are combined.
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