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Chapter 3: Problem 50

Give an example of an infinite group in which every nontrivial subgroup is infinite.

Short Answer

Expert verified
Provide an example of an infinite group where every nontrivial subgroup is infinite. The additive group of integers, denoted as \((\mathbb{Z},+)\), is an infinite group for which every nontrivial subgroup is infinite. In this group, the subgroup of all even integers, denoted as \(2\mathbb{Z}\), is an infinite nontrivial subgroup. Every other nontrivial subgroup of \((\mathbb{Z}, +)\) will also be infinite, as each subgroup must contain all integer multiples of its smallest positive integer, forming an infinite set.

Step by step solution

01

Choose an infinite group

Let's choose the additive group of integers, denoted as \((\mathbb{Z}, +)\), as our example of an infinite group.
02

Provide an example of a nontrivial subgroup

A nontrivial subgroup must not be the trivial group {0}. Let's choose the subgroup of all even integers, denoted as \(2\mathbb{Z}\), as our example of a nontrivial subgroup. This subgroup can be defined as \(2\mathbb{Z} = \{ ... -4, -2, 0, 2, 4, ...\}\).
03

Show that the nontrivial subgroup is infinite

To see that \(2\mathbb{Z}\) is infinite, consider that for any integer \(n\), we can find an even number (an element of \(2\mathbb{Z}\)) that is greater than \(n\). To do this, we can take \(n+1\) if \(n\) is odd, and \(n+2\) if \(n\) is even. Thus, there are always even integers beyond any integer \(n\), and so \(2\mathbb{Z}\) is an infinite subgroup.
04

Demonstrate how every nontrivial subgroup is infinite

Let \(H\) be any nontrivial subgroup of \((\mathbb{Z}, +)\). Since it is nontrivial, it must contain at least one nonzero integer. Let \(n\) be the smallest positive integer in \(H\). Due to the properties of subgroups, \(H\) must also contain all integer multiples of \(n\): \(kn\) for all \(k \in \mathbb{Z}\). This set is infinite, because, similar to step 3, for any integer \(m\), we can find a multiple of \(n\) that is greater than \(m\) by taking \(k=\lceil\frac{m}{n}\rceil+1\). Hence, H is an infinite subgroup.
05

Conclusion

We have shown that in the infinite additive group of integers \((\mathbb{Z}, +)\), every nontrivial subgroup is also infinite, meeting the criteria of the exercise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Additive Group of Integers
The concept of an additive group of integers is foundational in abstract algebra. It revolves around the set of all integers, which we denote as \(\textbf{Z}\), combined with the operation of addition. When we talk about the additive group of integers, we're referring to the mathematical structure that consists of the entire set of integers under the operation of standard addition.

The group has some specific properties that make it a valid group:
  • Closure: For any two integers \(a\) and \(b\), the sum \(a + b\) is also an integer.
  • Associativity: The equation \((a + b) + c = a + (b + c)\) holds for all integers \(a\), \(b\), and \(c\).
  • Identity element: The number 0 acts as an additive identity because when any integer is added to 0, it yields the same integer.
  • Inverse elements: For every integer \(a\), there exists an inverse \(-a\) such that \(a + (-a) = 0\).
Such characteristics are critical in many areas of mathematics, and understanding them is vital to grasp concepts in group theory.
Nontrivial Subgroup
A nontrivial subgroup of a group is one that is not simply the trivial group containing only the identity element. In the context of the additive group of integers, the identity element is 0, and thus the trivial subgroup is \({0}\).

A nontrivial subgroup must contain at least one other integer apart from the identity. For example, in the case of \(\textbf{Z}\), a subgroup like the even integers (denoted by \(2\textbf{Z}\)) is nontrivial. It includes all multiples of 2: \(... -4, -2, 0, 2, 4, ...\text{ etc.}\), which is clearly more than just the identity element. This concept is important because it introduces variability into the group structure and allows us to explore interesting aspects of group theory, like the classification of subgroups and their properties.
Subgroup Properties
Subgroups are smaller groups that live within a larger group and retain the group properties. To qualify as a subgroup, a subset \(H\) of a group \(G\) must satisfy specific criteria:
  • Closure within \(G\): For every pair of elements \(a, b\) in \(H\), the product \(a * b\) (in group operation terms) must also be in \(H\).
  • Contain the identity element of \(G\): In our case, since we're dealing with an additive group of integers, 0 must be in \(H\).
  • Contain inverses: For every element \(a\) in \(H\), its inverse \(-a\) must also be in \(H\).
  • Associativity: If the group operation is associative, which is a given for any group, then associativity naturally holds within the subgroup.
Understanding these properties helps to ensure that any structure we consider as a subgroup behaves consistently with the overarching group, thereby maintaining the integrity of its algebraic structure. In classroom settings, emphasizing these properties can significantly improve the comprehension of what defines a subgroup in group theory.

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Most popular questions from this chapter

Show that if \(G\) is a finite group of even order, then there is an \(a \in G\) such that \(a\) is not the identity and \(a^{2}=e\).

Let \(n=0,1,2, \ldots\) and \(n \mathbb{Z}=\\{n k: k \in \mathbb{Z}\\}\). Prove that \(n \mathbb{Z}\) is a subgroup of \(\mathbb{Z}\). Show that these subgroups are the only subgroups of \(\mathbb{Z}\).

Let \(a\) and \(b\) be elements in a group \(G\). Prove that \(a b^{n} a^{-1}=\left(a b a^{-1}\right)^{n}\) for \(n \in \mathbb{Z}\).

Let \(a\) and \(b\) be elements of a group \(G\). If \(a^{4} b=b a\) and \(a^{3}=e,\) prove that \(a b=b a\).

Describe the symmetries of a square and prove that the set of symmetries is a group. Give a Cayley table for the symmetries. How many ways can the vertices of a square be permuted? Is each permutation necessarily a symmetry of the square? The symmetry group of the square is denoted by \(D_{4}\).
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