Exploring Group Theory Concepts

Definition of a group.

• A group is a collection of numbers only.

• A group must have at least three elements.

• A group is a set with a binary operation satisfying closure, associativity, identity, and invertibility.

• A group is defined by its geometric properties.

What are the criteria for a subset to be a subgroup?

• A subset must contain at least two elements to be a subgroup.

• A subset is a subgroup if it is closed under the group operation only.

• A subset H is a subgroup if it contains the identity, is closed under the group operation, and is closed under inverses.

• A subset is a subgroup if it is non-empty and contains all group elements.

The concept of a group homomorphism.

• A group homomorphism is a type of group that contains subgroups.

• A group homomorphism is a function between two groups that preserves the group operation.

• A group homomorphism is a mapping that only applies to finite groups.

• A group homomorphism is a function that changes the group operation completely.

How do you solve the equation x * a = b in a group?

• x = b * a^(-1)

• x = b + a

• x = b / a

• x = a * b

What is a group action and how is it defined?

• A group action is a type of mathematical operation on numbers.

• A group action is a visual representation of group elements.

• A group action is a method of combining two groups into one.

• A group action is a function from a group G and a set X to X, denoted as G × X → X, satisfying specific properties.

Classify the group Z_6 and describe its structure.

• Z_6 is a cyclic group of order 6.

• Z_6 is a finite abelian group of order 4.

• Z_6 has an order of 12.

• Z_6 is a non-cyclic group of order 6.

What is the identity element in a group?

• The identity element is the element that can only be combined with itself.

• The identity element in a group is the element that leaves other elements unchanged when combined.

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

• The identity element is the element that always changes other elements.

Provide an example of a non-abelian group.

• Z3 (the cyclic group of order 3)

• A3 (the alternating group on 3 elements)

• D4 (the dihedral group of order 8)

• S3 (the symmetric group on 3 elements)

What is the kernel of a group homomorphism?

• The kernel is the set of elements that map to any element in the group.

• The kernel of a group homomorphism is the set of elements that map to the identity element of the codomain.

• The kernel is the entire group itself.

• The kernel consists of all elements in the codomain.

How can you determine if a group is cyclic?

• A group is cyclic if there exists an element such that every group element can be expressed as a power of that element.

• A group is cyclic if it has a finite number of elements.

• A group is cyclic if it is abelian.

• A group is cyclic if it contains at least two distinct elements.