Ultimate Abstract Quiz

Prop 1.1.1 (Groups)

Inverses are unique, identity is its own inverse, and cancellation works (a*x = a*y > x=y)

Prop 1.2.1 (Permutations)

Permutation composition forms a valid group (S_a)

Prop 1.2.2 (S_n Size & Commutativity)

Size is |S_n| = n!, and it is nonabelian if and only if n geq 3

Prop 1.2.3 (Cycles)

Elements stay in their cycles, and disjoint cycles commute

Prop 1.2.5 (Cycle Decomposition)

Every nontrivial permutation can be written as a unique product of disjoint cycles

Addition Groups (Z,Q,R under +):

Identity: 0, Inverse: -x, Rule: Infinite & Abelian (Commutative)

Multiplication Groups (Q^x, R^x under x):

Identity: 1, Inverse: 1/x(x^-1), Rule: You must drop 0 because 0 has no reciprocal

Symmetric Groups (S_n)

Identity: The “do-nothing” permutation, Inverse: Flip the cycles backwards, Rule: Nonabelian for n geq 3

What are the three group axioms?

Associativity, Identity Element, Inverse Element

What does it mean to say that an element of a group is nontrivial?

If (G, *) is a group and x E G is not the identity of G, we can say that x is a nontrivial element of G.

What does it mean to say that a group is abelian?

If (G, *) is a group and the binary operation is commutative, we can say that the group (G, *) is abelian (or commutative); otherwise, we can say that G is nonabelian (non-commutative).

What does it mean to say that a group is finite?

If (G, *) is a group and the set G contains finitely many elements, we can say that G is finite (or that G has finite order), we refer to the number of elements in G as the order of G, and we denote this number by |G|. If the group G contains infinitely many elements, we say that G is infinite (or that G has infinite order).

If A is a nonempty set, what does it mean to be a permutation of A?

Let A be a nonempty set. A bijection o: A > A is called a permutation of A.

What is the notation for the set of all permutations of A?

The set of all permutations of A is denoted by (S_a).

If n is a positive integer and A = {1,…,n}, what alternative notation do we use for the set of all permutations of A?

If n is a positive integer and A = {1,…,n}, then we use the notation S_n (instead of S_a) for the permutations of A, and we call (S_n, o) the symmetric group of degree n.

What is the identity permutation of a set?

S_a forms a group under composition of permutations of A. The identity of the group (S_a, o) is the identity permutation, and the inverse permutation is its inverse function.

What is the length of a cycle?

The positive integer d (where 1 leq d leq n) is called the length of a cycle sigma.

What does it mean to say that an integer belongs to a cycle?

The integer is one of the distinct elements x_0,…, x_d-1 that belongs to the cycle sigma.

What does it mean to say that two cycles are disjoint?

We say that cycles sigma=(x_0,…, x_d-1) and tau=(y_0,…, y_delta-1) are disjoint if the sets {x_0,…, x_d-1} and {y_0,…, y_delta-1} are disjoint.

Given a nontrivial permutation sigma, what is a cycle decomposition of sigma?

Fix a positive integer n, and suppose sigma E S_n is nontrivial. A cycle decomposition sigma is an expression of sigma as a composition of pairwise disjoint cycles, each having length greater than 1.