3) Order and other Group Theory Properties

Question 1

What is the order of a group

Answer 1

The cardinality of the set G

Question 2

What are the orders of the cyclic group Zn and the symmetric group Sn

Answer 2

|Zn| = n
|Sn| = n!

Question 3

What is the order of an element of a group

Answer 3

Question 4

What is the order of the identity element in a group

Answer 4

The identity element e has order 1, and this is the only element of order 1

Question 5

What can be said about the powers of an element with infinite order

Answer 5

Question 6

Describe the proof that all powers of an element of infinite order are distinct

Answer 6

Question 7

What type of subgroup does an element of infinite order generate

Answer 7

Infinite Cyclic Subgroup

Question 8

What are the properties of powers of an element of order n

Answer 8

Question 9

Describe the proof of properties of powers of an element of order n

Answer 9

Question 10

What type of subgroup is generated by an element of order n

Answer 10

Cyclic Subgroup of Order n

Question 11

If G is a cyclic group of order n, and m is a positive integer dividing n, show that G contains a subgroup of order m

Answer 11

Question 12

What can be said about the composition of any permutation

Answer 12

Every permutation is a product of pairwise disjoint cycles

Question 13

What is the relationship between disjoint cycles within a permutation

Answer 13

Question 14

What determines the order of a product of disjoint cycles in a permutation

Answer 14

The order of a product of disjoint cycles in a permutation is the l.c.m of the lengths of the individual cycles

Question 15

What defines a group theoretic property

Answer 15

A property P of groups is considered a group theoretic property if it is shared by all groups that are isomorphic to each other

Question 16

Give 3 examples of group theoretic properties

Answer 16

• Abelian Nature: This property is preserved under isomorphism because the operation structure that allows for commutativity is maintained between isomorphic groups.
• Order of the Group: If two groups are isomorphic, they have the same number of elements. Thus, the order of a group is invariant under isomorphism.
• Cyclic Nature: The property of being cyclic is maintained among isomorphic groups because the structural characteristic of having a generator element is preserved.

Question 17

Give 3 non-examples of group theoretic properties

Answer 17

• Being a group whose elements are matrices
• Being a group whose elements are numbers
• Being a group whose elements are permutations

Question 18

How can group theoretic properties be used to determine if two groups are not isomorphic

Answer 18

To prove that two groups are not isomorphic, one can identify a group theoretic property present in one group but absent in the other

Question 19

How are cyclic groups classified up to isomorphism according to their order

Answer 19