Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 13: Problem 1
The subgroup \(\langle 3\rangle\) of \(\mathbb{Z}\).
Short Answer
Expert verified
\( \langle 3 \rangle \) is the set of all multiples of 3 in \( \mathbb{Z} \).
Step by step solution
01
Understand the Group Z
The group \( \mathbb{Z} \) represents the set of all integers under the operation of addition. It includes elements like \(..., -3, -2, -1, 0, 1, 2, 3, ...\). In this group, every element is combined through addition.
02
Identify the Generator 3
The subgroup \( \langle 3 \rangle \) is generated by the integer 3. This means we consider all integer multiples of 3, that is ...., -6, -3, 0, 3, 6,... These form the subgroup within \( \mathbb{Z} \).
03
Describe the Subgroup \(\langle 3\rangle\)
The set \( \langle 3 \rangle = \{ 3n | n \in \mathbb{Z} \} \), where each element in the subgroup is an integer multiple of 3. This includes all numbers that can be written as \( 3 \times n \), where \( n \) is any integer.
04
Understand Properties of \(\langle 3\rangle\)
In the context of \( \mathbb{Z} \), \( \langle 3 \rangle \) is an infinite cyclic subgroup. The elements are closed under addition, meaning any sum or difference of elements in \( \langle 3 \rangle \) also belongs to \( \langle 3 \rangle \). Moreover, this subgroup is generated by adding or subtracting the generating element, 3, repeatedly.
05
Conclusion Regarding \(\langle 3\rangle\)
The subgroup \( \langle 3 \rangle \) includes all possible integer combinations obtained by adding or subtracting 3 any number of times. It does not include any integer that is not a multiple of 3, illustrating its identity defined by the generator 3.
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.
Cyclic Group
A cyclic group is a fundamental concept in group theory. It refers to a group that is generated by a single element, called the generator. All elements of a cyclic group can be written as powers of this generator. In mathematical terms, if \( g \) is a generator of a cyclic group \( G \), then every element of \( G \) can be expressed as \( g^n \) for some integer \( n \). This means that just by repeatedly applying the group operation to the generator, you can generate every element of the group.
- Cyclic groups can be finite or infinite. A finite cyclic group has a limited number of elements, while an infinite cyclic group goes on forever.
- The subgroup \( \langle 3 \rangle \), mentioned in the original exercise, is a classic example of an infinite cyclic group.
- Understanding cyclic groups is crucial, as many complex groups can be simplified down to combinations of cyclic groups.
Integer Multiples
In the context of the subgroup \( \langle 3 \rangle \), integer multiples are the building blocks. When we say a number is an integer multiple of another, it means it can be expressed as that number times an integer. Here, all numbers in the subgroup \( \langle 3 \rangle \) are integer multiples of 3, meaning they can be expressed as \( 3n \), where \( n \) is any integer.
- These integer multiples can be positive, negative, or zero. For example, -6, 0, 3 are all part of the subgroup \( \langle 3 \rangle \).
- This notion is essential in constructing other number-related mathematical structures, like integer lattices.
- The concept of integer multiples is also a stepping stone to understanding more advanced topics in algebra and number theory.
Group Theory
Group theory is a broad and impactful field of mathematics dealing with algebraic structures known as groups. Groups provide insight into symmetry, solutions to polynomials, and are fundamental in many areas of mathematics.
- A group consists of a set of elements and an operation that combines any two elements to form a third element within the same set. Here, the set of all integers \( \mathbb{Z} \) under addition forms a group.
- Group theory allows mathematicians to explore and develop the concepts of symmetry and invariance, which are applicable to physics, chemistry, and even computer science.
- By understanding groups like \( \langle 3 \rangle \), students can grasp broader concepts of group theory and explore its applications in various fields.
Infinite Subgroup
An infinite subgroup is a subgroup that contains an infinite number of elements. This often occurs in the context of larger, infinite groups. The subgroup \( \langle 3 \rangle \) is an infinite subgroup within the group of integers \( \mathbb{Z} \) since it includes all integer multiples of 3, such as ..., -9, -6, -3, 0, 3, 6, 9,...
- Infinite subgroups like \( \langle 3 \rangle \) have no upper bound on their number of elements.
- They are typically characterized by their generating element, which in this case is the number 3.
- Infinite subgroups are significant in mathematics because they serve as models for understanding infinity and the behavior of infinitely long sequences of numbers.
