permutation_group

introduction

%%visits: 2 ## intuition ## rigour :todo: Alternating groups are sort of symmetric, loosely speaking

Permuation group notation to cycle notation

:todo:

Transpositions

transpositions are just 2-cycles, and every cycle can be described as a 2-cycle. A great way to think of this is that you swap things together. {{file:./27_09_2021_10_39.svg}}

The figure above shows how to write $\begin{pmatrix} 1 &2&3&4 \\ 2&3&4&1 \end{pmatrix}$ in cycle-notation, which would be (1 2)(4 2)(2 3), just jot down which positions to swap, the order does not really matter. Now, an important point, We could change the order of the swaps and the notation looks completely different, but this represents the same permutation map, so think [[function]]s.

{{file:./figures/01_10_2021_10_44.svg}}

Alternating groups

What are alternating groups? Alternating groups are the set of all even permutations (that can be written as even 2-cycles)

exam clinic

resources

tags :math:groups_and_symmetries:

backlinks

• index_of_files.pd