MATH 470 - Exam 2 Verified Solutions
permutation ️a function from A to A that is both one-to-one and onto
permutation group ️a set of permutations of A that forms a group under function composition
Every permutation of a finite set can be written as... ️a cycle or as a product of disjoi...
permutation ✔️a function from A to A that is both one-to-one and onto
permutation group ✔️a set of permutations of A that forms a group under function composition
Every permutation of a finite set can be written as... ✔️a cycle or as a product of disjoint cycles
If the pair of cycles alpha = (a1, a2, ..., am) and beta = (b1, b2, ..., bn) have no entries in common
✔️Then alpha*beta = beta*alpha
The order of a permutation of a finite set written in disjoint cycle form is ✔️the least common multiple
of the lengths of the cycle
Sn ✔️Let A = {1, 2, ..., n} the set of all permutations of A is called the symmetric group of degree n and
is denoted by Sn
What is the order of Sn ✔️n!
If a permutation a can be expressed as a product of an even (odd) number of 2-cycles ✔️Then every
decomposition of a into a product of 2-cycles must have an even (odd) number of 2-cycles
Even permutation ✔️A permutation that can be expressed as a product of an even number of 2-cycles
A set of even permutations in Sn ✔️Forms a subgroup of Sn
An ✔️The group of even permutations of n symbols, the alternating group of degree n
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