Problem 17 - Algebra: Pure and Applied

Question

Give an example of a group with the indicated combination of properties: (a) an infinite cyclic group (b) an infinite Abelian group that is not cyclic (c) a finite cyclic group with exactly six generators (d) a finite Abelian group that is not cyclic

Step-by-Step Solution

Answer

(a) \( \mathbb{Z} \); (b) \( \mathbb{Q}^* \); (c) \( \mathbb{Z}_7 \); (d) \( \mathbb{Z}_2 \times \mathbb{Z}_2 \).

Step 1: Infinite Cyclic Group Example

An infinite cyclic group is a group generated by a single element where the group has infinitely many elements. The most common example of such a group is the group of integers under addition, denoted as \( \mathbb{Z} \). Every integer can be expressed as a product of a single generator, for example, 1 or -1.

Step 2: Infinite Abelian Group That is Not Cyclic

An infinite Abelian group that is not cyclic is one in which the group operation is commutative, but there is no single element whose powers can generate the whole group. A common example is the group \( \mathbb{Q}^* \), the multiplicative group of nonzero rational numbers, as it cannot be generated by a single rational number.

Step 3: Finite Cyclic Group with Exactly Six Generators

A finite cyclic group generated by an element \( g \) and having exactly six generators means \( |G| = 12 \) since the number of generators of a cyclic group of order \( n \) is \( \phi(n) \), the Euler's totient function. \( \phi(12) = 4 \) suggesting another revision. Proper example comes from a group of order 7, i.e., \( \mathbb{Z}_7 \), where 6 numbers are coprime with 7. \( \phi(7) = 6 \), therefore, 6 elements out of 7 will generate this cyclic group.

Step 4: Finite Abelian Group That is Not Cyclic

A finite Abelian group that is not cyclic must have more than one generator and can be expressed as a direct product of two or more groups. An example is \( \mathbb{Z}_2 \times \mathbb{Z}_2 \), which has 4 elements: \((0,0), (0,1), (1,0), (1,1)\). No single element generates all these elements.

Key Concepts

Infinite Cyclic Group

An infinite cyclic group is a group that is generated by only one element, yet contains an infinite number of elements. The quintessential example of such a group is the set of all integers under addition, denoted as \( \mathbb{Z} \). This group is quite intuitive and important in mathematics because every integer can be expressed by a multiple of a single generator.

For instance:

• If the generator is 1, then any integer \( n \) can be expressed as \( n \cdot 1 \).
• Using -1 as the generator, you can similarly represent every integer, albeit with signs reversed.

The ability to describe all elements of the group using a single generator is a defining feature of an infinite cyclic group, making it fundamental and straightforward at the same time.

Finite Abelian Group

A finite Abelian group is one where the group has a finite number of elements, and the operation in the group satisfies the commutative property, meaning the order of the operation does not matter. Such groups play a central role in fields like number theory and crystallography.

An example of a finite Abelian group that is not cyclic is the group \( \mathbb{Z}_2 \times \mathbb{Z}_2 \), consisting of pairs of elements that operate under addition modulo 2. This group:

• Contains 4 elements: \( (0,0), (0,1), (1,0), (1,1) \).
• Operates such that addition of elements is commutative, meaning \( (a,b) + (c,d) = (c,d) + (a,b) \).
• Is not cyclic because no single element can generate every other element through repeated operations.

Cyclic Group Generators

Generators in cyclic groups are the building blocks that allow the construction of an entire group by repeated application of a group operation. Whenever you have a cyclic group, you know that there exists at least one generator.

Take, for example, the cyclic group \( \mathbb{Z}_7 \) of integers modulo 7. This group:

• Contains generators that are coprime with 7; these numbers are 1, 2, 3, 4, 5, 6.
• Has these 6 generators because \( \phi(7) = 6 \), where \( \phi \) is Euler's totient function, which counts the integers up to a given integer \( n \) that are coprime with \( n \).
• Means that any element \( g \), when used in repeated operations like additions or multiplications, can generate the entire set of elements within the group.

Abelian Group Properties

Abelian groups, named after the mathematician Niels Henrik Abel, are characterized by the property that their group operation is commutative. This property simplifies analysis and understanding because the order in which you perform operations does not affect the result.

Some key properties of Abelian groups include:

• Closure: The result of the operation on any two elements of the group is also in the group.
• Associativity: Changing the grouping of the operations does not change the result.
• Identity Element: There exists an element in the group which, when used in the group operation with any element of the group, leaves the other element unchanged.
• Inverses: For every element, there exists another element that combines with it to yield the identity element.

For example, in the setting of \( \mathbb{Z}_2 \times \mathbb{Z}_2 \), these properties hold true, ensuring that no matter how elements are selected or arranged, the group operation produces consistent and predictable results.