Problem 12 - Abstract Algebra: Theory and Applications

Question

Let \(N\) be a normal subgroup of \(G\). If \(N\) and \(G / N\) are solvable groups, show that \(G\) is also a solvable group.

Step-by-Step Solution

Answer

Question: Prove that if \(N\) is a normal subgroup of \(G\) such that both \(N\) and \(G/N\) are solvable groups, then \(G\) is also a solvable group. Answer: We construct a subnormal series for \(G\) by combining the subnormal series of \(N\) and \(G/N\). This is done by first listing the subnormal series of \(N\) and then appending the preimages in \(G\) of the subgroups in the subnormal series of \(G/N\). We then show that this series satisfies the definition of a solvable group, thus concluding that \(G\) is solvable.

Step 1: Recall the definition of solvable groups

A group \(G\) is said to be solvable if there is a sequence of subgroups, known as a subnormal series, such that \(G_0 = G\), \(G_n = \{ e \}\) (the trivial group), and each \(G_i\) is a normal subgroup of \(G_{i-1}\) with \(G_{i-1}/G_i\) being an abelian group for \(i = 1, 2, \dots, n\).

Step 2: Utilize given information

We are given that \(N\) is a normal subgroup of \(G\), and that both \(N\) and the quotient group \(G/N\) are solvable. This means that we have subnormal series for both \(N\) and \(G/N\), say: - For \(N\): \(N_0 = N, N_1, N_2, \dots, N_n = \{ e \}\) where each \(N_i \trianglelefteq N_{i-1}\) and \(N_{i-1}/N_i\) is abelian. - For \(G/N\): \((G/N)_0 = G/N, (G/N)_1, (G/N)_2, \dots, (G/N)_m = \{eN\}\) where each \((G/N)_i \trianglelefteq (G/N)_{i-1}\) and \((G/N)_{i-1}/(G/N)_i\) is abelian.

Step 3: Construct a subnormal series for G

We will now construct a subnormal series for G by combining the subnormal series of \(N\) and \(G/N\). This can be done as follows: 1. Start with the subnormal series of \(N\): \(G_0 = N_0 = N, G_1 = N_1, G_2 = N_2, \dots, G_n = N_n = \{ e \}\). 2. Next, let \(H_i\) be the preimage of the subgroup \((G/N)_{i-1}\) in \(G\) for \(i = n + 1, n + 2, \dots, n + m\). In other words, \(H_i = (G/N)_{i-1}^{-1}(N)\). This means that \(H_i\) is a subgroup of \(G\), and it satisfies the relation \(H_i/N = (G/N)_{i-1}\). 3. Add this preimage to our series: \(G_{n+1} = H_1, G_{n+2} = H_2, \dots, G_{n+m} = H_m = G\).

Step 4: Show that G has a solvable series

We now have a subnormal series for G: \(G_0, G_1, \dots, G_n, G_{n+1}, \dots, G_{n+m}\). We need to show that this series satisfies the definition of a solvable group. For \(i = 1, 2, \dots, n\), we already know that \(G_i \trianglelefteq G_{i-1}\) and \(G_{i-1}/G_i\) is abelian because it is the subnormal series of \(N\). For \(i = n + 1, n + 2, \dots, n + m\), we have that \(H_i/N \trianglelefteq (G/N)_{i-1} \trianglelefteq (G/N)_{i-2}\), which implies that \(H_i \trianglelefteq H_{i-1}\) in \(G\). Moreover, since \(H_i/N = (G/N)_{i-1}\) and \((G/N)_{i-1}/(G/N)_i\) is abelian, we can conclude that \(H_{i-1}/H_i\) is abelian. Therefore, \(G_i \trianglelefteq G_{i-1}\) and \(G_{i-1}/G_i\) is abelian for \(i = n + 1, n + 2, \dots, n + m\). Thus, we have constructed a subnormal series for \(G\), making \(G\) solvable.

Key Concepts

Normal Subgroup

In group theory, a normal subgroup plays a central role. A subgroup \( N \) of a group \( G \) is called a normal subgroup if it is invariant under conjugation by members of \( G \).
What this means is that for every element \( g \) in \( G \) and every element \( n \) in \( N \), the element \( gng^{-1} \) is still in \( N \).
Some properties of normal subgroups include the ability to form quotient groups and a key position in the definitions of solvable groups and other algebraic structures. Often denoted as \( N \trianglelefteq G \), normal subgroups allow for the partitioning of \( G \) into cosets, which have important implications for understanding the structure and behaviors of groups.
Understanding normal subgroups is essential when exploring the compositional structure of larger groups and when assessing the solvability of those groups.

Quotient Group

A quotient group is formed by dividing a group \( G \) by one of its normal subgroups \( N \). It is often denoted as \( G/N \), and is a way of simplifying the structure of \( G \) while retaining some of its essential properties.
In essence, we form a new group where each element is a coset of \( N \) in \( G \). The operation on these cosets is defined as:\[ (gN)(hN) = (gh)N \]
Key aspects of quotient groups include:

• They help in analyzing the properties of the original group in a simplified fashion.
• If \( G \) is a finite group, then the order (size) of each coset and of \( G/N \) can provide insights into divisibility and symmetry structure.
• Quotient groups are crucial when investigating solvable groups, as they relate to the series construction that leads to such a group's classification.

Understanding how quotient groups function is vital when examining the decomposition of structures in mathematics.

Subnormal Series

A subnormal series is a sequence of subgroups starting from a group \( G \) and descending to the trivial subgroup, each one being normal in the previous one.
This concept is crucial in the study of solvable groups, where at each step, the quotient of two successive groups in the sequence must be an abelian group.
Some characteristics of subnormal series include:

• A given sequence: \( G = G_0 \geq G_1 \geq \cdots \geq G_n = \{ e \} \).
• Every subgroup in the sequence being a normal subgroup of the one before it.
• The quotients \( G_{i-1}/G_i \) being significantly simpler, often abelian.

The formation of a subnormal series helps in unraveling the complex structure of groups, providing a "stepping-down" method to reach a particularly simple group form, namely the trivial group.

Abelian Group

An Abelian group is defined by its commutative property where the group operation \(*\) satisfies \( a * b = b * a \) for all elements \( a \) and \( b \) in the group.
Named after the mathematician Niels Henrik Abel, these groups are among the simplest forms of groups in abstract algebra.
Key aspects of Abelian groups:

• All cyclic groups are Abelian, which means any group generated by a single element is inherently commutative.
• They serve as the building blocks for more complex group structures, often present in subnormal series for solvable groups.
• Understanding Abelian groups is fundamental, as they form the backbone for more advanced concepts in group theory, such as modules and vector spaces.

Recognizing the properties of Abelian groups helps in breaking down complex problem-solving steps into straightforward calculations supported by well-known rules.