Permutations Flashcards
What is a permutation in the context of group theory?
A permutation is a rearrangement of the elements of a set.
True or False: A group is a set equipped with an operation that satisfies closure, associativity, identity, and invertibility.
True
What is the symmetric group on n elements denoted as?
It is denoted as S_n.
Fill in the blank: The number of permutations of n distinct objects is _____ .
n!
Define a cyclic permutation.
A cyclic permutation is a permutation where a group of elements is rotated among themselves.
What is the order of a permutation?
The order of a permutation is the smallest positive integer k such that applying the permutation k times returns to the original arrangement.
True or False: Every permutation can be expressed as a product of transpositions.
True
What is a transposition?
A transposition is a permutation that swaps two elements and leaves all others unchanged.
What does it mean for a permutation to be even?
A permutation is even if it can be expressed as a product of an even number of transpositions.
What does it mean for a permutation to be odd?
A permutation is odd if it can be expressed as a product of an odd number of transpositions.
What is the notation for the alternating group of degree n?
It is denoted as A_n.
How many elements are in the symmetric group S_n?
S_n has n! elements.
What is the relationship between S_n and A_n?
A_n is a subgroup of S_n consisting of all even permutations.
Define the concept of the cycle structure of a permutation.
The cycle structure is the decomposition of a permutation into disjoint cycles.
What is a k-cycle?
A k-cycle is a cycle that permutes k elements while fixing all other elements.
What is the order of a permutation given its cycle structure?
The order is the least common multiple of the lengths of the cycles in the cycle structure.
True or False: The identity permutation is an element of every symmetric group.
True
What is the effect of a permutation on the elements of a set?
The permutation rearranges the elements of the set.
Fill in the blank: The inverse of a permutation is a permutation that _____ the original permutation.
reverses
What is a permutation group?
A permutation group is a group whose elements are permutations of a given set.
What is the symmetric group S_3?
S_3 is the group of all permutations of three elements.
How many elements does A_3 have?
A_3 has 3 elements.
What is the Cayley’s theorem in the context of permutation groups?
Cayley’s theorem states that every group is isomorphic to a subgroup of a symmetric group.
What is the stabilizer of an element in a group action?
The stabilizer is the set of permutations that leave that element unchanged.
Define a group action.
A group action is a way of describing how a group interacts with a set.
What is a coset in group theory?
A coset is a form of partitioning a group formed by multiplying all elements of a subgroup by a fixed group element.
What is the orbit of an element under a group action?
The orbit is the set of elements that can be reached by applying the group elements to that element.
Fill in the blank: The kernel of a group action is the set of all elements that _____ every element of the set.
fix
True or False: The symmetric group S_n is non-abelian for n ≥ 3.
True
What is a complete group in terms of permutations?
A complete group is a group where every automorphism is inner.
What is the significance of the sign of a permutation?
The sign indicates whether the permutation is even or odd.
What does it mean for two permutations to be conjugate?
Two permutations are conjugate if one can be transformed into the other by a change of basis in the symmetric group.
What is a representation of a permutation group?
A representation is a homomorphism from the group to the general linear group of some vector space.
Fill in the blank: A permutation can be represented by a _____ on the set of its elements.
matrix
What is the significance of the orbit-stabilizer theorem?
It relates the size of the orbit of an element to the size of its stabilizer.
True or False: The alternating group A_n is simple for n ≥ 5.
True
What is the relationship between the symmetric group and combinatorial objects?
The symmetric group acts on combinatorial objects such as permutations and combinations.
Name one application of permutation groups in modern mathematics.
One application is in the study of algebraic structures, such as Galois theory.
