2) Subgroups and Homomorphisms Flashcards

1
Q

What is the trivial subgroup

A

In any group G with identity element e, the set {e} is a trivial subgroup of G

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2
Q

What is a proper subgroup

A

A subgroup H of G is called proper if H ≠ G

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3
Q

How does the identity element of a subgroup relate to the identity element of the entire group

A

In any subgroup H of a group G, the identity element of H is the same as the identity element of G

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4
Q

What is the subgroup criteria

A

A non-empty subset H of G is a subgroup if and only if the following two conditions both hold -
(i) ∀g, h ∈ H we have gh ∈ H,
(ii) ∀h ∈ H we have h^-1 ∈ H

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5
Q

Are the natural numbers a subgroup of the integers

A

N is not a subgroup of Z. Although the sum of two natural numbers is a natural number, the inverse n of n ∈ N is not in N

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6
Q

What can be said about the intersection of two subgroups within the same group

A

The intersection of any two subgroups of the same group is itself a subgroup

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7
Q

Give an example of a group G and subgroups H ≤ G, K ≤ G
such that the union H ∪ K is not a subgroup of G

A
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8
Q

Give an example of a subgroup of a non-abelian group that is abelian

A
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9
Q

What is the special linear group

A

SL(n, K) = {A ∈ GL(n, K) | det A = 1}
is a subgroup of GL(n, K)

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10
Q

What are the different types of subgroups of matrices commonly studied in linear algebra and group theory

A

UT(n, K) - Upper Uni-Triangular Matrices
T(n,K) - Upper Triangular Matrices
SL(n, K) - Special Linear Group
D(n, K) - Invertible Diagonal Matrices
Scal(n,K) - Scalar Matrices

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11
Q

What are the powers of a group element

A
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12
Q

What does < g > denote

A

{g^k | k ∈ Z}

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13
Q

What are the properties of powers of group elements

A
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14
Q

What type of subgroup is formed by all powers of a fixed element

A

Cyclic Subgroup

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15
Q

Describe the proof that a cyclic subgroup is an abelian subgroup

A
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16
Q

What is a cyclic subgroup

A

Let G be a group and a ∈ G. The subgroup < a > of
G is called the cyclic subgroup generated by a

17
Q

When is a group cyclic

A

A group G is called cyclic, if there exists an element
a ∈ G such that G = < a >

18
Q

Prove that (Q, +) is not a cyclic group

A
19
Q

What is a generator

A
  • An element from which every other element of the group can be derived through repeated applications of the group operation
  • In cyclic groups, a single generator can produce the entire group
20
Q

What is a group homomorphism

A
21
Q

What is the relationship between isomorphisms and homomorphisms

A

All isomorphisms are homomorphisms, but not all homomorphisms are isomorphisms

22
Q

What is the image and kernel of a homomorphism

A
23
Q

What are the properties of homomorphisms

A
24
Q

Describe the proof of the properties of homomorphisms

A
25
Q

What can be said about the inverse of an isomorphism

A
26
Q

Describe the proof that the inverse of an isomorphism is also an isomorphism

A