3) Order and other Group Theory Properties Flashcards

1
Q

What is the order of a group

A

The cardinality of the set G

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2
Q

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

A

|Zn| = n
|Sn| = n!

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3
Q

What is the order of an element of a group

A
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4
Q

What is the order of the identity element in a group

A

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

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5
Q

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

A
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6
Q

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

A
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7
Q

What type of subgroup does an element of infinite order generate

A

Infinite Cyclic Subgroup

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8
Q

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

A
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9
Q

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

A
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10
Q

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

A

Cyclic Subgroup of Order n

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11
Q

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

A
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12
Q

What can be said about the composition of any permutation

A

Every permutation is a product of pairwise disjoint cycles

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13
Q

What is the relationship between disjoint cycles within a permutation

A
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14
Q

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

A

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

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15
Q

What defines a group theoretic property

A

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

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16
Q

Give 3 examples of group theoretic properties

A
  • 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.
17
Q

Give 3 non-examples of group theoretic properties

A
  • Being a group whose elements are matrices
  • Being a group whose elements are numbers
  • Being a group whose elements are permutations
18
Q

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

A

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

19
Q

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

A