Given a [finite set](Finite%20sets.md) $n$ distinguishable objects, we refer to any rearrangement of these objects as a _permutation._ A _permutation_ is an ordering of elements in a [sequence](Sequences.md). We may denote this as a re-ordering of a sequence $(j_1,j_2...,j_n)$ to the sequence $(i_1,i_2...,i_n)$ denoted as $P_{i_1,i_2...,i_n}^{j_1,j_2...,j_n}.$ # Properties of Permutations 1) Taking the [[Signum]] of a given permutation $P$, $\mathrm{sgn}(P)=+1$ if $P$ is even and $\mathrm{sgn}(P)=-1$ if $P$ is odd. # The permutation group --- # Recommended reading For an introduction to permutations that motivates the introduction of the [Levi-Civita Symbol](Levi-Civita%20Symbol.md) see: * [Altland, A. von Delft, J. Mathematics for Physicists, Cambridge University Press, 2019](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md) pgs. 10-12. %%This note does not really belong in set theory. I think also permutations are hard to explain, but they reveal something about exchange symmetries.%% #MathematicalFoundations/SetTheory #MathematicalFoundations/Algebra #MathematicalFoundations/Numbers