Reading...
Play button
Play button
Use LEFT and RIGHT arrow keys to navigate between flashcards;
Use UP and DOWN arrow keys to flip the card;
H to show hint;
A reads text to speech;
19 Cards in this Set
- Front
- Back
|
Permutation of a set A
|
A function from A to A that is one-to-one and onto.
|
|
|
Permutation group of a set A
|
A set of permutations of A that forms a group under function composition.
|
|
|
Symmetric group of order n, denoted Sn
|
The set of all permutations of a set A.
|
|
|
Products of disjoint cycles
|
Every permutation of a finite set can be written as a cycle or as a product of disjoint cycles.
|
|
|
Order of a permutation
|
The order of a permutation of a finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles.
|
|
|
Always even or always odd
|
If a permutation α can be expressed as a product of an even number of 2-cycles, then every decomposition of α into a product 2-cycles has an even number of 2-cycles. A similar statement holds for odd.
|
|
|
Even (odd) permutation
|
A permutation that can be decomposed into a product of an even (odd) number of 2-cycles.
|
|
|
Alternating group of degree n, denoted An
|
The group of even permutations of Sn.
|
|
|
Order of An
|
For n > 1, the order of An is n!/2.
|
|
|
Product of 2-cycles
|
Every permutation is Sn, n > 1, is a product of 2-cycles.
|
|
|
(1)
|
φ carries the identity of G to the identity of .
|
|
|
(2)
|
For every integer n and for every element a of G, .
|
|
|
(3)
|
For elements a and b in G, a and b commute if and only if φ(a) and φ(b) commute.
|
|
|
(4)
|
|a| = |φ(a)| for all a in G (isomorphisms preserve order).
|
|
|
(5)
|
For a fixed integer k and a fixed group element b in G, xk = b has the same number of solutions in G as does the equation xk = φ(b) in .
|
|
|
(1)
|
G is Abelian if and only if is Abelian.
|
|
|
(2)
|
G is cyclic if and only if is cyclic.
|
|
|
(3)
|
φ-1 is an isomorphism from onto G.
|
|
|
(4)
|
If K ≤ G, then ≤ .
|
