Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 10
F眉r reelle Zahlen \(a, b\) sei \(t_{a, b}: \mathbb{R} \rightarrow \mathbb{R}\) definiert durch \(t_{a, b}(x)=a x+b .\) Es sei \(G:=\left\\{t_{a, b} \mid a, b \in \mathbb{R}, a \neq 0\right\\} .\) Zeigen Sie: (a) Die Menge \(G\) bildet mit der Komposition von Abbildungen eine Gruppe. (b) Es ist \(N:=\left\\{t_{1, b} \mid b \in \mathbb{R}\right\\}\) Normalteiler in \(G\). (c) Es gilt \(G / N \cong \mathbb{R} \backslash\\{0\\}\).
Short Answer
Expert verified
(a) G is a group under composition. (b) N is normal in G. (c) G/N is isomorphic to \(\mathbb{R} \setminus \{0\}\).
Step by step solution
01
Define the composition operation
The operation we need to consider for the set \(G\) is the composition of functions. For functions \(t_{a,b}\) and \(t_{c,d}\), their composition is given by: \(t_{a,b} \circ t_{c,d}(x) = a(cx + d) + b = acx + (ad + b).\) Hence, the composition of two functions in \(G\) yields another function of the same form, ensuring the operation is closed.
02
Identity element in G
To form a group, \(G\) must have an identity element under composition. The identity function \(t_{1,0}(x) = x\) serves this role, as it satisfies \(t_{1,0} \circ t_{a,b} = t_{a,b} \circ t_{1,0} = t_{a,b}\) for all \(t_{a,b} \in G\).
03
Inverse element in G
Each element \(t_{a,b} \in G\) must have an inverse. The inverse of \(t_{a,b}\) is \(t_{\frac{1}{a},-\frac{b}{a}}\), so that \(t_{a,b} \circ t_{\frac{1}{a},-\frac{b}{a}} = t_{1,0}\). Thus, every element in \(G\) has an inverse under composition.
04
Associativity of function composition
Function composition is always associative, meaning for any functions \(t_{a,b}, t_{c,d}, t_{e,f} \in G\), it holds that \((t_{a,b} \circ t_{c,d}) \circ t_{e,f} = t_{a,b} \circ (t_{c,d} \circ t_{e,f})\).
05
Conclude G is a group
Since the set \(G\) with composition has closure, an identity element, inverses, and is associative, \(G\) forms a group.
06
Definition and normality of N
The set \(N=\{t_{1,b} \mid b \in \mathbb{R}\}\) consists of translations. We check whether it is a normal subgroup by ensuring that for any \(t_{a,b} \in G\) and \(t_{1,c} \in N\), the element \(t_{a,b} \circ t_{1,c} \circ t_{a,b}^{-1}\) also belongs to \(N\). Calculation shows \(t_{a,b} \circ t_{1,c} \circ t_{a,b}^{-1}(x)= t_{1,c}\), confirming normality.
07
Determine the quotient group G/N
Since \(N\) is normal, we consider elements \([t_{a,b}]\) in the quotient group \(G/N\). The map \(\phi: G/N \rightarrow \mathbb{R} \setminus\{0\}\) given by \([t_{a,b}] \mapsto a\) is well-defined and an isomorphism, as it preserves the group operation and all non-zero real numbers are covered.
08
Conclude G/N is isomorphic to \(\mathbb{R} \setminus \{0\}\)
Therefore, the quotient group \(G/N\) is isomorphic to the multiplicative group of non-zero real numbers, \(\mathbb{R} \setminus\{0\}\), completing the proof.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Composition
In group theory, function composition is a fundamental operation when dealing with transformations or mappings.
It involves combining two functions to form a new function. For functions such as \(t_{a,b}(x) = ax + b\) and \(t_{c,d}(x) = cx + d\), composition is achieved by substituting the output of one function into the input of the other.
Mathematically, composing \(t_{a,b}\) with \(t_{c,d}\) results in \(t_{a,b} \circ t_{c,d}(x) = a(cx + d) + b = acx + (ad + b)\).
Key points about composition in group \(G\):
It involves combining two functions to form a new function. For functions such as \(t_{a,b}(x) = ax + b\) and \(t_{c,d}(x) = cx + d\), composition is achieved by substituting the output of one function into the input of the other.
Mathematically, composing \(t_{a,b}\) with \(t_{c,d}\) results in \(t_{a,b} \circ t_{c,d}(x) = a(cx + d) + b = acx + (ad + b)\).
Key points about composition in group \(G\):
- **Closure:** The composition of two functions from \(G\) results in another function of the same form, keeping the set closed under this operation.
- **Associativity:** Function composition is inherently associative, which means the order of applying brackets does not matter.
- **Identity:** The function \(t_{1,0}(x) = x\) serves as the identity function, leaving any function in \(G\) unchanged when composed.
- **Inverses:** Every function \(t_{a,b}\) has an inverse, \(t_{\frac{1}{a}, -\frac{b}{a}}\), satisfying \(t_{a,b} \circ t_{\frac{1}{a}, -\frac{b}{a}} = t_{1,0}\).
Normal Subgroup
A normal subgroup is a special subset of a group that holds some extra properties, which affect how the group can be partitioned into cosets. In the context of the exercise, \(N := \{t_{1,b} \mid b \in \mathbb{R}\}\) is specified as a normal subgroup of \(G\).
This means that for all elements \(t_{a,b} \in G\) and \(t_{1,c} \in N\), the element \(t_{a,b} \circ t_{1,c} \circ t_{a,b}^{-1}\) remains in \(N\).
Some essential characteristics include:
This means that for all elements \(t_{a,b} \in G\) and \(t_{1,c} \in N\), the element \(t_{a,b} \circ t_{1,c} \circ t_{a,b}^{-1}\) remains in \(N\).
Some essential characteristics include:
- **Normality Condition:** Group symmetry ensures left and right cosets coincide. For any element in \(G\), when conjugated by an element of \(N\), it yields another element of \(N\).
- **Importance:** Normal subgroups allow the group to be broken into simpler, more manageable pieces, facilitating the formation of quotient groups.
- **Structure:** In our specific case, \(N\) consists of all translations, represented as \(t_{1,b}\).
Quotient Group
The quotient group is a concept that arises when a group \(G\) is divided by one of its normal subgroups \(N\). It provides insight into the structure and characteristics of \(G\).
More formally, \(G/N\) represents the set of all distinct cosets of \(N\) in \(G\). In our problem, \(N = \{t_{1,b} \mid b \in \mathbb{R}\}\), making the quotient group \(G/N\).
Main aspects include:
More formally, \(G/N\) represents the set of all distinct cosets of \(N\) in \(G\). In our problem, \(N = \{t_{1,b} \mid b \in \mathbb{R}\}\), making the quotient group \(G/N\).
Main aspects include:
- **Cosets:** Each coset is a set formed by multiplying a fixed element from \(G\) by every element in \(N\). They represent the distinct classes in the quotient group, \([t_{a,b}]\).
- **Group Operation:** The operation of the quotient group is defined by combining these cosets as usual, using the operation that the group \(G\) originally employed.
- **Output Group:** For this exercise, \(G/N\) is isomorphic to \(\mathbb{R} \setminus \{0\}\), indicating that the structure of the original group \(G\) can, essentially, "collapse" into this simpler, more understandable group.
Isomorphism
Isomorphism is a concept in group theory denoting a strong type of symmetry between groups, implying the groups have the same structure.
An isomorphism is a bijective map between two groups that maintains the group operations' consistency within the map.
In our context, the map \(\phi: G/N \rightarrow \mathbb{R} \setminus \{0\}\) defined by \([t_{a,b}] \mapsto a\) establishes an isomorphism.
Key characteristics:
An isomorphism is a bijective map between two groups that maintains the group operations' consistency within the map.
In our context, the map \(\phi: G/N \rightarrow \mathbb{R} \setminus \{0\}\) defined by \([t_{a,b}] \mapsto a\) establishes an isomorphism.
Key characteristics:
- **Bijective Mapping:** The map \(\phi\) is both injective (one-to-one) and surjective (onto), ensuring it reaches all elements in the target group unique from each in the source group.
- **Operation Preservation:** The map keeps the group operation unchanged. The group combination in \(G/N\) translates perfectly into multiplication within \(\mathbb{R} \setminus \{0\}\).
- **Structural Insight:** Through isomorphism, we realize that \(G/N\) and \(\mathbb{R} \setminus \{0\}\) are, fundamentally, the same group presented in different "clothes."
