Question

Zeigen Sie: Jede Untergruppe von \((\mathbf{Z},+)\) ist von der Form \((n \mathbf{Z},+)\).

Short answer

Every subgroup of \((\mathbf{Z},+)\) is \((n\mathbf{Z},+)\) for some integer \(n\).

Step-by-step solution

Understanding the Problem

We are asked to show that every subgroup of \((\mathbf{Z},+)\) has the form \((n\mathbf{Z},+)\). This means we have to prove that any subgroup of the integers under addition can be described by all integer multiples of some integer \(n\).

Identifying Key Concepts

A subgroup \(H\) of \((\mathbf{Z},+)\) must be closed under addition and contain the additive inverse of each of its elements. Moreover, it must contain the identity element, 0.

Considering Subgroups of \((\mathbf{Z},+)\)

Consider an arbitrary subgroup \(H\) of \(\mathbf{Z}\). If \(H = \{0\}\), it can be described as \((0\mathbf{Z},+)\). If \(H\) contains nonzero elements, let \(d\) be the smallest positive integer in \(H\).

Proving Closure under Addition

If \(d \in H\), by the subgroup property, all integer multiples of \(d\) (i.e., \(n \times d\) for any integer \(n\)) must also be in \(H\). Thus, \((d\mathbf{Z},+) \subseteq H\).

Showing Minimal Generator

Suppose \(H\) contains an element \(h\) not in \(d\mathbf{Z}\), then \(h = qd + r\) where \(0 \leq r < d\), by the Division Algorithm. As \(h, qd \in H\), it follows that \(r = h - qd \in H\), contradicting \(d\)'s minimality unless \(r = 0\). Therefore \(H = (d\mathbf{Z},+)\).

Conclusion

Every subgroup \(H\) of \((\mathbf{Z},+)\) can be expressed as \((n\mathbf{Z},+)\), where \(n\) is the smallest positive integer in \(H\), confirming the form \((n\mathbf{Z},+)\) for each subgroup.

Key concepts

Group Theory

Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group consists of a set equipped with an operation that combines any two of its elements to form a third element. This operation must satisfy four essential properties: closure, associativity, identity, and invertibility.

• Closure: If you take any two elements from the group and use the operation on them, the result must also be an element in the group.
• Associativity: The operation must be associative, meaning that the way in which the operation is performed does not change the result. In formulaic terms, for any elements \(a, b, c\) in the group, \((a * b) * c = a * (b * c)\).
• Identity: There must be an element in the group that is the identity for the operation. For any element \(a\) in the group, \(a * e = a\) and \(e * a = a\), where \(e\) is the identity element.
• Invertibility: For each element in the group, there must be another element in the group that, when used with the operation, results in the identity element.

These rules make groups a central concept in algebra, and they play a fundamental role in understanding the structure of various mathematical systems.

Integer Addition

Integer addition is a simple operation where integers are combined following the basic arithmetic addition operation. In the context of group theory, integers (\(\mathbf{Z}\)) form a group under addition.

• This means that when you add any two integers, the result is also an integer. This showcases the property of closure.
• The operation is associative, meaning \((a + b) + c = a + (b + c)\) for any integers \(a, b,\) and \(c\).
• The integer 0 acts as the identity element since \(a + 0 = a\) for any integer \(a\).
• Every integer \(a\) has an additive inverse, \(-a\), such that \(a + (-a) = 0\).

Under these properties, the set of integers \(\mathbf{Z}\) forms an infinite group with the operation of addition.

Minimal Generators

In group theory, a generator is an element that can be used to express all elements of a group by repeatedly applying the group operation. A minimal generator of a subgroup of integers refers to the smallest positive integer that generates the entire subgroup under addition.
If a subgroup \(H\) of integers \(\mathbf{Z}\) has a positive integer \(d\) as a generator, it means that every element of \(H\) can be written as \(nd\) for some integer \(n\). Thus, \(d\) is the minimal generator if it is the smallest positive integer that meets this condition.

• It guarantees that all multiples of \(d\) are within the subgroup \(H\).
• If any integer smaller than \(d\) could also serve as a generator, then \(d\) would not be minimal.

This concept is vital for understanding the structure of subgroups within the integers, allowing a simple and unified way to describe every subgroup.

Subgroup Properties

Subgroups are subsets of a group that themselves form a group under the same operation. For a set to be a subgroup, it must satisfy specific subgroup criteria:

• Closure: The subgroup must be closed under the group operation. If \(a\) and \(b\) are in the subgroup, their operation result \(a + b\) should also be in the subgroup.
• Identity Element: The subgroup must include the identity element of the larger group.
• Inverse Elements: Every element in the subgroup must have an inverse also within the subgroup.

In the group of integers under addition, every subgroup can be expressed in the form \((n\mathbf{Z},+)\), meaning they are multiples of a particular integer \(n\). This integer \(n\) is the minimal generator, and it ensures the subgroup's closure and invertibility, as all integer multiples naturally meet these conditions. This elegant description simplifies working with infinite subsets like the integers in mathematical proofs and analysis.