Understanding Cyclic Groups

What is a cyclic group?

• A group that cannot be generated by any element

• A group that is always finite

• A group that can be generated by a single element

• A group that can be generated by multiple elements

In additive notation, what is the identity element?

• -1

• 0

• x

• 1

What must a subgroup containing an element 'x' also contain?

• The inverse of 'x' and the identity element

• All elements of the group

• None of the above

• Only the element 'x'

Which of the following is an example of an infinite cyclic group?

• The integers under addition

• The integers mod 3

• The integers mod 4

• The trivial group

What is a classic example of a finite cyclic group?

• The complex numbers under addition

• The real numbers under multiplication

• The integers mod N under addition

• The integers under addition

Why are finite cyclic groups called 'cyclic'?

• Because they repeat their elements in a cycle

• Because they have infinite elements

• Because they are not closed under the group operation

• Because they are generated by multiple elements

What is the complete collection of cyclic groups?

• The real numbers and the complex numbers

• The integers mod 2 and mod 3 only

• All finite groups

• The integers and the integers mod N

What does the Fundamental Theorem of Finitely Generated Abelian Groups state?

• Only finite groups can be cyclic

• Any abelian group can be broken into cyclic groups

• Every group is cyclic

• Cyclic groups are not important

What are the fundamental building blocks for finitely generated abelian groups?

• Non-abelian groups

• Cyclic groups

• Infinite groups

• Trivial groups

What is the significance of cyclic groups in abstract algebra?

• They are not significant

• They are the basis for all groups

• They are the building blocks for finitely generated abelian groups

• They are only used in number theory