Groups

Question 1

What does it mean when a property is closed?

Answer 1

An operator * is closed in a set S if for every pair of

elements a and b in S the quantity a*b ∈ S

Question 2

What does it mean when a property is Associative?

Answer 2

An operator * is associative in a set S if for every three
elements a, b and c in S
(ab)c = a(bc)

Question 3

What does it mean when a property has an identity?

Answer 3

An operator * has an identity in a set S if there is an

element e in S such that for every element a in S ae = ea = a

Question 4

What does it mean when a property has an inverse?

Answer 4

An element a in S has an inverse a-1 for an operator * if

aa-1 =a-1a=e

Question 5

What makes a set a group?

Answer 5

1) Closed (includes all the elements in the universe)
2) Associative (symmetry down the diagonal)
3) Identity element (e.g ea =a , ae = a)
4) Inverse element (e.g a + (-a) =0)