Understanding Subgroups and Lagrange's Theorem

What is the primary purpose of studying subgroups within a group G?

• To find the largest possible group

• To understand the structure of G by examining its smaller parts

• To determine the number of elements in G

• To identify the identity element of G

According to Lagrange's Theorem, what must be true about the order of a subgroup H of a finite group G?

• The order of H is always greater than the order of G

• The order of H is equal to the order of G

• The order of H is a divisor of the order of G

• The order of H is always a prime number

If a group G has 323 elements, what are the possible orders of its subgroups according to Lagrange's Theorem?

• 17, 19, 323

• 1, 323

• 1, 2, 3, 323

• 1, 17, 19, 323

What does Lagrange's Theorem not guarantee about subgroups of a given order?

• That the subgroup is unique

• That the subgroup is trivial

• That the subgroup is abelian

• That a subgroup of that order exists

In the example of the group A4, which order does not have a corresponding subgroup despite being a divisor of the group's order?

• 3

• 6

• 4

• 2

What is the key observation about left cosets in the proof of Lagrange's Theorem?

• They are larger than the subgroup

• They contain the identity element

• They are all the same size and do not overlap

• They always overlap with each other

What is the significance of the identity element in the context of subgroups and cosets?

• It is present in every subgroup but not in cosets

• It is not present in any subgroup

• It is present in every subgroup and coset

• It is only present in the trivial subgroup

What is the term used for the number of cosets when partitioning a group G?

• Index

• Rank

• Degree

• Order

How does the proof of Lagrange's Theorem differ for abelian and non-abelian groups?

• It does not differ; the proof is the same

• Non-abelian groups do not have cosets

• Only abelian groups can be partitioned into cosets

• Left and right cosets are the same for abelian groups but may differ for non-abelian groups

What is the main takeaway from the video regarding the use of cosets in group theory?

• Cosets are only useful for infinite groups

• Cosets help in proving the existence of subgroups

• Cosets are used to partition groups into non-overlapping sets

• Cosets are irrelevant to Lagrange's Theorem