abstract algebra iii

Remembering: What is the second 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 Sylow p-subgroup to be normal in a group?

• The Sylow p-subgroup is the only subgroup of its order
• The Sylow p-subgroup is unique and has p elements
• Every element of the group normalizes the Sylow p-subgroup
• The Sylow p-subgroup is the unique subgroup of its order and has p elements

Applying: If a group has 90 elements and is known to have a Sylow 3-subgroup, what can be said about the number of elements in the Sylow 3-subgroup?

• It has 3 elements
• It has 9 elements
• It has 27 elements
• It has 81 elements

Analyzing: How does the second 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 second part of Sylow's theorem be used to determine if a group is simple?

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

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

• No, the second part of Sylow's theorem only deals with the normality of Sylow p-subgroups
• Yes, if every Sylow p-subgroup is normal
• No, the second part of Sylow's theorem does not address the abelian property
• Yes, if there is only one normal Sylow p-subgroup for every prime number p