abstract algebra ii

Remembering: What is the first part of Sylow's Theorem?

• There exists a Sylow p-subgroup for every prime number p
• Every Sylow p-subgroup is normal
• The number of Sylow p-subgroups is congruent to 1 mod p
• The number of Sylow p-subgroups divides the order of the group

Understanding: What does it mean for a subgroup to be a Sylow p-subgroup?

• The subgroup is normal and has p elements
• The subgroup has p elements and is the only subgroup of its order
• The subgroup has p elements and is the unique subgroup of its order
• The subgroup is normal and is the unique subgroup of its order

Applying: If a group has 60 elements and is known to have a Sylow 2-subgroup, what can be said about the number of elements in the Sylow 2-subgroup?

• It has 2 elements
• It has 4 elements
• It has 8 elements
• It has 16 elements

Analyzing: How does the first part of Sylow's theorem help to classify groups?

• By determining if a group is abelian
• By determining the number of normal subgroups in a group
• By determining the number of Sylow p-subgroups in a group
• By determining the order of a group

Evaluating: Can the first part of Sylow's theorem be used to determine if a group is simple?

• Yes, if there is only one Sylow p-subgroup for every prime number p
• No, the first part of Sylow's theorem does not address simplicity
• Yes, if the number of Sylow p-subgroups is equal to the order of the group
• No, the first part of Sylow's theorem only deals with the existence of Sylow p-subgroups

Creating: Can the first part of Sylow's theorem be used to determine if a group is abelian?

• Yes, if there is only one Sylow p-subgroup for every prime number p
• No, the first part of Sylow's theorem does not address abelian property
• No, the first part of Sylow's theorem only deals with the existence of Sylow p-subgroups
• Yes, if the number of Sylow p-subgroups is equal to the order of the group