Chapter 1 and 2

Question 1

Binary operation (on a set G)

Answer 1

A function that assigns to each ordered pair of elements of G an element of G.

Question 2

Operation table, a.k.a. Cayley table

Answer 2

A table of results of a binary operation (written in the form of a multiplication table).

Question 3

Closure

Answer 3

The condition that members of an ordered pair from a set G combine to yield a member of G.

Question 4

Group

Answer 4

A set G together with a binary operation for which the following properties are satisfied:

(1) Associativity: (ab)c = a(bc) for all a, b, c in G.
(2) Identity: there exists an element e in G for which ae = ea = a for all a in G.
(3) Inverses: For each element a in G, there is an element b in G for which ab = ba = e.

Question 5

Commutative, a.k.a. Abelian (pertaining to a group G)

Answer 5

Having the property that ab = ba for all a and b in G.

Question 6

Uniqueness of the Identity

Answer 6

A group contains only one identity element.

Question 7

Cancellation

Answer 7

In a group, left and right cancellation hold.

Left: ab = ac implies b = c.
Right: ba = ca implies b = c.

Question 8

Uniqueness of inverses

Answer 8

For each element a in a group G, there is a unique element b in G for which ab = ba = e.