Binary Operations and Group Theory

Question 1

What is a Binary Operation?

Answer 1

A Binary operation * on a set S is a rule that assigns the element x*y to any ordered pair of elements x, y in S. Binary operation may or not be closed, commutative or associative.

Question 2

What does it mean for integers to be congruent?

Answer 2

The integers x and y are said to be congruent modulo m id the difference between them is a multiple of m.

Question 3

What is a way of representing a binary operation?

Answer 3

A Cayley table

Question 4

What is a Group?

Answer 4

A group is defined for a closed, associative binary operation, where an identity element exists and each element has an inverse

Question 5

What is an Abelian Group?

Answer 5

If the operation is commutative, the group is called abelian

Question 6

What is the period of an element?

Answer 6

The period of an element x of a group is the smallest non-negative interger n such that xⁿ=e (where e is the identity)

Question 7

What is a cyclic group?

Answer 7

G is a cyclic group is gⁿ=e for some generator g.

Question 8

What is the order of the symmetry group of an n-sided polygon?

Answer 8

2n

Question 9

What is a Subgroup?

Answer 9

any subset of G that gives rise to a group under the operation * is referred to as a subgroup of {G, * }

Question 10

What is Lagrange’s Theorem?

Answer 10

Lagrange’s Theorem states that the order of a subgroup of a finite group is a factor of the order of the group

Question 11

What does the term isomorphic mean (in group theory)?

Answer 11

Two groups are isomorphic if they have the same structure

Question 12

Give three properties of isomorphic graphs

Answer 12

For isomorphic graphs G and H:
G is abelian if and only if H is abelian.
the periods of the elements of G are the same as those of the elements of H.
the orders of the subgroups of G are the same as those of H.

Question 13

Give three results for isomorphisms?

Answer 13

All cyclic groups of a particular order are isomorphic to each other
all groups of order 4 are isomorphic to either the cyclic group if the Klein 4-group
there are two distinct groups of order 6: cyclic groups and groups isomorphic to D₃

Question 14

What is a non-trivial subgroup?

Answer 14

A non-trivial subgroup is any subgroup other than {e}

Question 15

What is a generator of a group?

Answer 15

It is said that g is a generator of the Group G (of order n) if gⁿ=e, and no smaller powers of g equals e.

Question 16

What is identity element?

Answer 16

If there is an element e ∈ S, such that e * a = a * e = a, for all a ∈ S, then e is said to be an identity element for the binary operation *.

Question 17

What is an inverse of an element

Answer 17

If, for an element a ∈ S, there exists an element b ∈ , such that a * b = b * a = e, then b is said to be an inverse of a ( and a is an inverse of b)

Question 18

What does it mean for an operation to be closed?

Answer 18

x * y must be an element of S for the set S and operation *.

Question 19

What does commutative mean?

Answer 19

A binary operation * on S is said to be commutative if a * b = b * a fr all a,b ∈ S

Question 20

What does associative mean?

Answer 20

A binary operation * on S is said to be associative if (a * b) * c = a * (b * c), for all a,b,c ∈ S

Question 21

What is a proper subgroup?

Answer 21

A proper subgroup is any subgroup apart from the whole group.