Normal Subgroups and Group Theory Quiz

Which of the following best defines a normal subgroup?

• A subgroup that is not closed under the group operation

• A subgroup that is equal to its conjugates in the group

• A subgroup that has no identity element

• A subgroup that is equal to the entire group

What is a proper normal subgroup?

• A subgroup that contains all the elements of the group

• A normal subgroup that is equal to the group itself

• A normal subgroup that is neither the trivial subgroup nor the group itself

• A subgroup that does not contain the identity element

Which of the following is true about the improper normal subgroup?

• It contains no elements

• It is equal to the identity subgroup

• It is the same as the whole group

• It is always finite

Which of the following best describes the Hamilton group?

• A non-abelian group in which every subgroup is normal

• A group where no normal subgroups exist

• An abelian group with no identity element

• A cyclic group with infinite order

Which of the following is a necessary condition for a subgroup to be a normal subgroup?

• It must be finite

• It must have more elements than the group

• Its left and right cosets must be equal

• It must contain only the identity element

What can be said about the intersection of two normal subgroups?

• It is never a normal subgroup

• It is always the trivial subgroup

• It is also a normal subgroup

• It is equal to the union of the two subgroups

Why is a subgroup of index 2 always normal?

• Because it has only one element

• Because there are only two cosets, which must be equal in both left and right forms

• Because it is not a proper subgroup

• Because it is always abelian

What is a simple group?

• A group with only one element

• A group that contains no normal subgroups other than the trivial subgroup and the group itself

• A group with infinitely many normal subgroups

• A group that is always cyclic

What is a quotient group?

• A group formed by removing all elements of a subgroup

• A group formed by dividing a group into cosets of a normal subgroup

• A subgroup that is equal to the group itself

• A subgroup that contains only one element

Which condition must be satisfied for a quotient group to exist?

• The subgroup must be improper

• The subgroup must be infinite

• The subgroup must be normal

• The subgroup must be simple