Problem 8
Question
Suppose that \(G\) is a finitely generated abelian group every element of which, except the identity, has infinite order. Show that \(G \cong \mathbb{Z}^{s}\), where \(s\) is defined by the property that \(G\) is generated by \(s\) elements, but is not generated by \(s-1\) elements.
Step-by-Step Solution
Verified Answer
G is isomorphic to \( \mathbb{Z}^{s} \).
1Step 1: Recognize the Structure of G
Since every element of the abelian group \( G \), except the identity, has infinite order, it implies that there are no torsion elements other than the identity. In the context of finitely generated abelian groups, this means \( G \) is a free abelian group.
2Step 2: Apply Fundamental Theorem for Finitely Generated Abelian Groups
The Fundamental Theorem for Finitely Generated Abelian Groups states that any such group can be expressed as the direct sum of cyclic groups. Given that \( G \) has no torsion, this means \( G \) can only be decomposed into infinite cyclic groups, specifically \( \mathbb{Z} \).
Key Concepts
Finitely Generated Abelian GroupsInfinite Cyclic GroupsFree Abelian Groups
Finitely Generated Abelian Groups
Finitely generated abelian groups are an important topic in algebra. These groups are built from a combination of simple structures. Specifically, an abelian group is one where the group operation is commutative. This means that for any two elements, say \( a \) and \( b \), in the group, it holds that \( a + b = b + a \).
A finitely generated abelian group refers to a group that can be generated by a finite set of elements. These groups can be expressed as a combination of cyclic subgroups. The foundational result concerning these groups is the Fundamental Theorem for Finitely Generated Abelian Groups. This theorem tells us that such a group can be decomposed into a direct sum of cyclic groups. These cyclic groups might be finite (i.e., have elements of finite order) or infinite.
A finitely generated abelian group refers to a group that can be generated by a finite set of elements. These groups can be expressed as a combination of cyclic subgroups. The foundational result concerning these groups is the Fundamental Theorem for Finitely Generated Abelian Groups. This theorem tells us that such a group can be decomposed into a direct sum of cyclic groups. These cyclic groups might be finite (i.e., have elements of finite order) or infinite.
- If every element except the identity has infinite order, as in the problem statement, then these groups are free from torsion. This means each cyclic component is infinite.
- An example decomposition for a torsion-free finitely generated abelian group could be \( \mathbb{Z}^n \) where each component in the direct sum is an infinite cyclic group.
Infinite Cyclic Groups
Infinite cyclic groups are a key building block in algebraic structures. Imagine a single element, \( g \), and the group consists of all integer powers of \( g \). Such a group is infinite since you can multiply or divide \( g \) infinitely many times by itself, using positive, negative, or zero exponents.
The group of integers \( \mathbb{Z} \) under addition is a classic example of an infinite cyclic group. Here, the number 1 generates the entire group because you can reach any integer by adding or subtracting 1 as needed.
The group of integers \( \mathbb{Z} \) under addition is a classic example of an infinite cyclic group. Here, the number 1 generates the entire group because you can reach any integer by adding or subtracting 1 as needed.
- For any integer \( n \), \( n \) can be expressed as \( 1 + 1 + ... + 1 \) (n times) or by subtracting when n is negative.
- Infinite cyclic groups play a role in understanding the structure of larger, more complex groups. Any finitely generated abelian group free from torsion elements can be seen as a direct sum of several infinite cyclic groups.
Free Abelian Groups
A free abelian group feels like the algebraic version of freedom in terms of generation. Whereas not every group is free, those that fall under this category mimic infinitely stretchable extensions of simpler components. In particular, a free abelian group is one that has a maximal linearly independent set of generators. This means no generator can be expressed as a combination of the others.
Given the set of generators, these groups can be described as specially-dimensioned grids on an integer lattice. For example, \( \mathbb{Z}^2 \) is a free abelian group because you can think of it as the entire 2D plane lattice formed by integer coordinates.
Given the set of generators, these groups can be described as specially-dimensioned grids on an integer lattice. For example, \( \mathbb{Z}^2 \) is a free abelian group because you can think of it as the entire 2D plane lattice formed by integer coordinates.
- In the context of finitely generated, torsion-free abelian groups, the free nature simplifies the group's structure to that of multiple direct sums of infinite cyclic groups.
- These groups have a rank, which corresponds to how many free factors they break into and thus, relates directly to the integer lattice's dimensionality.
Other exercises in this chapter
Problem 5
Suppose that \(a\) and \(b\) are positive integers with highest common factor \(d\). Show that $$ \mathbb{Z}_{a} \times \mathbb{Z}_{b} \cong \mathbb{Z}_{d} \tim
View solution Problem 7
Suppose that \(G\) is an abelian group. Show that the set \(T\) of elements of finite order is a subgroup of \(G\) and that every element of \(G / T\), except t
View solution Problem 9
Suppose that \(G\) is a finitely generated abelian group. Show that \(G \cong \mathbb{Z}^{s} \times T\), where \(T\) is a finite group.
View solution Problem 11
Suppose that \(p\) and \(q\) are primes and that \(p
View solution