0
votes
1answer
30 views

Formalizing a permutation as a bijection between the same set?

The standard definition I usually find regarding permutations is a bijection from a set to itself, in other words: $\text{a function }f:\{ 1, 2,\dots,n \} \mapsto \{ 1, 2,\dots,n \} \text{ which is a ...
1
vote
1answer
18 views

Permutation cycles in Jacobson's Basic Algebra I.

Nathan Jacobson's Basic Algebra I Second Edition, Section 1.6 Cycle Decompositions of Permutations, page 51, exercise 4 says: Show that if $\alpha$ is any permutation then $$\alpha (i_1 i_2 \cdots ...
1
vote
0answers
38 views

Notation for number of tensor permutations

I have a tensor (or set if you will) that consists of N elements, and each element has a limited number of values it can take. For example: ...
5
votes
1answer
74 views

Is there a name for this given type of matrix?

Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$? (The motivation for this ...
1
vote
3answers
218 views

Notation: permutation and its inverse

Consider the sequence $S = (A, B, C, D, E)$ and the permutation $\pi = (4, 1, 3, 5, 2)$: Which of the following is true? $$ \pi(S) = (B, E, C, A, D) \quad and \quad \pi^{-1}(S) = (D, A, C, E, B) ...
1
vote
0answers
37 views

Notation for Restriction of Permutation

Suppose $\sigma$ and $\tau$ are permutations such that $\sigma(x)\not=x\implies \sigma(x)=\tau(x)$. Intuitively, I would like to think of $\sigma$ as a restriction (or projection) of $\tau$ onto a ...
1
vote
2answers
123 views

confusion over transpositions and cycle notation

Suppose we have a permutation on the set {1,2,3,4,5} and we express it in the cycle notation as (2,5,3). I interpret this to mean that every time we apply the permutation, 2 gets sent to 5, 5 gets ...
2
votes
1answer
96 views

Cycle notation question

With $\alpha = (12345)$ in the cycle notation, I should interpret it as: $1\mapsto 2 \mapsto 3\mapsto 4\mapsto 5\mapsto 1$ I need to find out $\alpha^2$ and write it in cyclic notation. As I am not ...
4
votes
4answers
482 views

How to read permutation symbols like $(123)$?

I'd be grateful for some help reading permutation symbols such as $(123)$. Does it mean, when applied to a target sequence such as $(x y z w)$, "replace the element in the first slot of the target ...