Exploring Group Theory

What are the four properties that define a group?

• Closure, Associativity, Identity Element, Invertibility

• Distributivity

• Commutativity

• Associative Property

Given the set of integers under addition, is it a group? Why or why not?

• No, the set of integers under addition is not a group because it lacks an identity element.

• No, because the set of integers does not include inverses for all elements.

• Yes, the set of integers under addition is a group.

• Yes, but only for positive integers under addition.

If G is a group and a, b are elements of G, what is the result of the operation a * b?

• a * b is an element of G.

• a * b is equal to b * a.

• a * b is an element of G but not unique.

• a * b is not defined in G.

Explain the concept of the identity element in a group.

• The identity element is the largest element in the group.

• The identity element is the element that can be combined with itself to produce any other element.

• The identity element is the element that has no effect when combined with itself.

• The identity element in a group is an element that, when combined with any element of the group, does not change that element.

What is the inverse of an element in a group? Provide an example.

• In the group of natural numbers under addition, the inverse of 3 is 0.

• In the group of real numbers under addition, the inverse of 3 is 3.

• In the group of integers under multiplication, the inverse of 3 is 1/3.

• For example, in the group of integers under addition, the inverse of 3 is -3, since 3 + (-3) = 0, where 0 is the identity element.

How do you determine if a subset of a group is also a group?

• A subset H of a group G is a group if it has at least one element.

• A subset H of a group G is a group if it is finite in size.

• A subset H of a group G is a group if it satisfies closure, contains the identity element, and includes inverses for all its elements.

• A subset H of a group G is a group if it contains all elements of G.

What is a cyclic group? Give an example.

• An example of a cyclic group is the group of integers under addition, denoted as (Z, +), where any integer can be generated by adding 1 (or -1) repeatedly.

• The group of rational numbers under multiplication, denoted as (Q, *)

• The group of non-zero real numbers under addition, denoted as (R*, +)

• The set of all even integers under addition, denoted as (2Z, +)

Define a subgroup and provide a criterion for a subset to be a subgroup.

• A subgroup is a subset of a group that contains all elements of the group.

• A subset H is a subgroup if it is empty and contains the identity element.

• A subgroup is a subset of a group that is also a group. A subset H is a subgroup if it is non-empty, closed under the group operation, and closed under taking inverses.

• A subgroup must include at least half of the elements of the group.

What is the order of a group? How is it determined?

• The order of a group is the sum of its elements.

• The order of a group is determined by its subgroup structure.

• The order of a group is the number of its elements.

• The order of a group is its highest element.

Explain the concept of isomorphism in group theory.

• Isomorphism refers to the process of combining two groups into one larger group.

• Isomorphism is a structural equivalence between two groups, indicating they have the same group structure.

• Isomorphism is a method for measuring the size of a group.

• Isomorphism indicates that two groups have different elements but the same number of elements.