Problem 1 - A Book of Abstract Algebra

Question

As a provisional definition, let us call a finite abelian group "decomposable" if there are elements \(a_{1}, \ldots, a_{n} \in G\) such that: (D1) For every \(x \in G\), there are integers \(k_{1}, \ldots, k_{n}\) such that \(x=a_{1}^{k_{1}} a_{2}^{k_{2}} \cdots a_{n}^{k_{n}}\). (D2) If there are integers \(l_{1}, \ldots, l_{n}\) such that \(a_{1}^{l_{1}} a_{2}^{l_{2}} \cdots a_{n}^{l_{n}}=e\) then \(a_{1}^{l_{1}}=a_{2}^{t_{2}}=\cdots\) \(=a_{n}^{l_{n}}=e\) If (D1) and (D2) hold, we will write \(G=\left[a_{1}, a_{2}, \ldots, a_{n}\right]\). Let \(G^{\prime}\) be the set of all products \(a_{2}^{l_{2}} \cdots a_{n}^{l_{8}}\), as \(l_{2}, \ldots, l_{n}\) range over \(\mathbb{Z}\). Prove that \(G^{\prime}\) is a subgroup of \(G\), and \(G^{\prime}=\left[a_{2}, \ldots, a_{n}\right]\).

Step-by-Step Solution

Answer

\(G'\) is a subgroup of \(G\), and \(G' = [a_2, \ldots, a_n]\) is decomposable.

Step 1: Understand what needs to be proved

We need to show two things: First, that \(G'\), the set consisting of elements \(a_2^{l_2} \cdots a_n^{l_n}\), forms a subgroup of \(G\); second, that \(G'\) can be expressed as \([a_2, \ldots, a_n]\), meaning it satisfies the conditions \(\text{(D1)}\) and \(\text{(D2)}\) for a decomposable group.

Step 2: Verify subgroup criteria for G'

A subset \(H\) of a group \(G\) is a subgroup if it satisfies: (1) the identity element is in \(H\), (2) if \(x, y \in H\), then \(xy \in H\), and (3) if \(x \in H\), then \(x^{-1} \in H\). Check these for \(G'\).

Step 3: Check identity element in G'

The identity element \(e\) of \(G\) can be expressed as \(e = a_2^0 \cdots a_n^0\), showing that \(e \in G'\).

Step 4: Verify closure under operation in G'

For any \(x = a_2^{m_2} \cdots a_n^{m_n}\) and \(y = a_2^{l_2} \cdots a_n^{l_n}\) in \(G'\), their product \(xy = a_2^{m_2+l_2} \cdots a_n^{m_n+l_n}\) is also in \(G'\) by the closure of integers under addition.

Step 5: Verify inverse property in G'

For any \(x = a_2^{m_2} \cdots a_n^{m_n}\) in \(G'\), its inverse is \(x^{-1} = a_2^{-m_2} \cdots a_n^{-m_n}\), which is also a product of the same generators with integer exponents, so \(x^{-1} \in G'\).

Step 6: Establish G' is decomposable satisfying (D1) and (D2)

For \(\text{(D1)}\), any element \(a_2^{l_2} \cdots a_n^{l_n}\) is trivially expressible from \(a_2, \ldots, a_n\). For \(\text{(D2)}\), if \(a_2^{l_2} \cdots a_n^{l_n} = e\), then each \(a_i^{l_i} = e\) since \(G\) is abelian and not reducible, confirming \(G' = [a_2, \ldots, a_n]\) satisfies both conditions.

Key Concepts

Group Theory

Group Theory is an area of mathematics that studies the algebraic structures known as groups. A group is a set equipped with an operation that combines any two of its elements to form another element from the same set.
In group theory, one of the key properties of a group is the presence of an identity element and inverses for every element. This makes groups essential in understanding symmetry, as they encapsulate the mathematical rules behind transformations.
Groups can be finite or infinite, but our focus here is on finite Abelian groups, which are groups that are both finite in number and commutative, meaning the order in which we combine elements doesn't matter. This commutativity and structure are particularly useful for decomposing groups into simpler, smaller subgroups.

Decomposable Group

A decomposable group is a type of finite Abelian group that can be 'split' into simpler components, or generators, such that any element in the group can be expressed as a combination of these generators.
The idea of a decomposable group revolves around two conditions, (D1) and (D2):

• (D1) Every element can be represented as a product of power of its generators.
• (D2) An equation of the product that results in the identity implies that each component must equal the identity.

This decomposition is significant because it simplifies complex group operations by reducing them to operations on the generators. It allows us to understand the group's structure by examining the interactions between its generators.

Subgroup Criteria

A subgroup is a subset of a group that itself satisfies the group properties. For a subset to qualify as a subgroup, it must meet specific criteria:

• Include the identity element of the parent group.
• Be closed under the group operation, meaning the result of combining any two elements of the subset must also be in the subset.
• Contain the inverses of its elements, ensuring that for any element in the subset, its inverse is also included.

In the exercise, we verify these for the subgroup formed by the elements of the form \(a_2^{l_2} \cdots a_n^{l_n}\), called \(G'\). Each criterion ensures this subset behaves like a group on its own within the larger group structure, maintaining the rules necessary for group operations.

Identity Element

The identity element is a fundamental concept in group theory, representing a unique element in a group that leaves other elements unchanged when combined with them. It acts as a 'do-nothing' operation.
For an Abelian group, the identity element is often denoted as \(e\). In the context of the exercise, when we say \(e = a_2^0 \cdots a_n^0\), it emphasizes that the identity element can be represented as the product of zero powers of any element, which would equal the identity, or as a trivial product of the generators.
The inclusion of the identity element in a subgroup is crucial for ensuring that the subgroup retains the structure of a group.

Inverse Property

The inverse property states that for every element in a group, there exists an inverse such that when they are combined under the group operation, the identity element is the result.
In simple terms, an element \(x\) has an inverse \(x^{-1}\) where \(x \cdot x^{-1} = e\). In our exercise, we ensure the subgroup \(G'\) contains the inverses of its elements, satisfying the subgroup criteria.
To find the inverse, you typically take the negative power of the element, as demonstrated: \(x^{-1} = a_2^{-m_2} \cdots a_n^{-m_n}\). This property ensures that we can 'undo' any operation within the group, keeping the group well-rounded and ensuring that all operations are reversible.