3) Order and other Group Theory Properties Flashcards
What is the order of a group
The cardinality of the set G
What are the orders of the cyclic group Zn and the symmetric group Sn
|Zn| = n
|Sn| = n!
What is the order of an element of a group
What is the order of the identity element in a group
The identity element e has order 1, and this is the only element of order 1
What can be said about the powers of an element with infinite order
Describe the proof that all powers of an element of infinite order are distinct
What type of subgroup does an element of infinite order generate
Infinite Cyclic Subgroup
What are the properties of powers of an element of order n
Describe the proof of properties of powers of an element of order n
What type of subgroup is generated by an element of order n
Cyclic Subgroup of Order n
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
What can be said about the composition of any permutation
Every permutation is a product of pairwise disjoint cycles
What is the relationship between disjoint cycles within a permutation
What determines the order of a product of disjoint cycles in a permutation
The order of a product of disjoint cycles in a permutation is the l.c.m of the lengths of the individual cycles
What defines a group theoretic property
A property P of groups is considered a group theoretic property if it is shared by all groups that are isomorphic to each other
Give 3 examples of group theoretic properties
- 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.
Give 3 non-examples of group theoretic properties
- Being a group whose elements are matrices
- Being a group whose elements are numbers
- Being a group whose elements are permutations
How can group theoretic properties be used to determine if two groups are not isomorphic
To prove that two groups are not isomorphic, one can identify a group theoretic property present in one group but absent in the other
How are cyclic groups classified up to isomorphism according to their order
