Oxford University Mathematical Institute
Linear Algebra II for Mathematical Moderations
Lecture 1 (Tuesday 18 January 2011): Permutations
Note. These skeletal notes are intended as a sort of ‘handout’. They are
designed to be printed off to form a template for your personal notes. A full text
will be published after the first few lectures have taken place.
Definition. A permutation of a set Sis a bijective (one-to-one and onto)
map S→S. The set of all permutations of Swill be denoted Sym(S) or Sym S.
Throughout these lectures the set Swill be finite, usually with nmembers. If
S={1,2, . . . , n}, or if it does not matter what set of size nis under consideration
we write Sym (n) instead of Sym(S) or SymS. Many authors use Snor Σn.
Cauchy’s two-line notation for permutations.
Notation: the image of x∈Sunder the map ρ:S→Swill be written xρ .
Composition (multiplication) of permutations.
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