Problem 4 - A Book of Abstract Algebra

Question

The center of a group \(G\) is the normal subgroup \(C\) of \(G\) consisting of all those elements of \(G\) which commute with every element of \(G\). Suppose the quotient group \(G / C\) is a cyclic group; say it is generated by the element \(C a\) of \(G / C\). Prove each of the following: Conclude that if \(G / C\) is cyclic, then \(G\) is abelian.

Step-by-Step Solution

Answer

If \(G / C\) is cyclic, then every element in \(G\) commutes, making \(G\) abelian.

Step 1: Understanding the Definitions

We are given that the center of a group \(G\) is the normal subgroup \(C\) of \(G\), consisting of all elements of \(G\) that commute with every element of \(G\). This means for any element \(c \in C\), \(cg = gc\) for all \(g \in G\). We also know \(G / C\) is a cyclic group generated by \(Ca\) for some \(a \in G\).

Step 2: Expressing Elements of Quotient Group

Since \(G / C\) is generated by \(Ca\), every element of \(G / C\) can be written as \((Ca)^n\) for some integer \(n\). This implies each element of \(G\) is of the form \(a^n c\) where \(c \in C\).

Step 3: Proving Commutativity

To prove \(G\) is abelian, we need to show that any two elements in \(G\) commute. Consider two arbitrary elements \(g = a^m c_1\) and \(h = a^n c_2\) of \(G\) where \(c_1, c_2 \in C\). Then, their product is \(gh = (a^m c_1)(a^n c_2) = a^m a^n c_1 c_2\).

Step 4: Using Properties of the Center and Cyclic Group

Since elements of the center \(C\) commute with all elements of \(G\), \(c_1 a^n = a^n c_1\) and \(c_2 a^m = a^m c_2\). Thus, \(gh = a^{m+n} c_1 c_2 = a^{m+n} c_2 c_1\).

Step 5: Concluding Commutativity of Elements

Similarly, we find \(hg = a^n a^m c_2 c_1 = a^{n+m} c_2 c_1\). Since \(m + n = n + m\) in integers, we have \(gh = hg\). Hence every pair of elements in \(G\) commute.

Step 6: Final Conclusion

From the above arguments, since every pair of elements in \(G\) commute, \(G\) is an abelian group.

Key Concepts

Cyclic Quotient Group

In group theory, a quotient group (or factor group) is a mathematical concept that involves partitioning a group, denoted as \( G \), by a normal subgroup \( C \). When the quotient group \( G/C \) is cyclic, it means that there exists an element in \( G/C \), often expressed as \( Ca \), that can generate every other element in the group by powering, which is multiplying it by itself repeatedly. For instance, if \( G/C \) is cyclic and generated by \( Ca \), then every element in \( G/C \) can be written in the form \((Ca)^n\), with \( n \) being an integer. This characteristic implies a certain regular structure within the group, as all elements are derived from a single source or generator. Cyclic groups are fundamentally simple to understand as they are essentially just repeated applications of the group operation on the generator. This cyclic nature plays a crucial role in determining other structural properties of \( G \).
This generator concept simplifies the understanding of the group's structure, allowing the connection to be made that if \( G/C \) is cyclic, \( G \) may exhibit more straightforward behaviors, such as being an abelian group.

Normal Subgroup

A normal subgroup \( C \) of a group \( G \) is a subgroup where any element \( c \in C \) commutes with all elements of \( G \) if conjugated. That means for any \( g \in G \), the element \( g^{-1}cg \) will still lie within \( C \). The center of a group, \( C \), is a particular instance of a normal subgroup.

• It contains elements that commute with every element in \( G \).
• If you take any \( c \in C \) and any \( g \in G \), then you have \( cg = gc \).

Thus, normality ensures a well-defined operation in forming quotient groups such as \( G/C \). Such structures are essential for simplifying complex groups into more comprehensible smaller ones, offering insight into group properties. Normal subgroups simplify the analysis of the group's internal symmetries and help in classifying groups and their behaviors. They also serve as milestones in group homomorphisms and play a vital role in higher mathematical structures.

Abelian Group

An abelian group is a group where every pair of elements commutes. This means for any two elements \( g \) and \( h \) in the group, their product is independent of the order they are multiplied in, i.e., \( gh = hg \). In our context, proving some

• that if \( G/C \) is cyclic, then \( G \) is abelian.
• Elements of the form \( a^n c \) (for any \( c \in C \)) preserve commutative properties when multiplied.

Because each \( c \in C \) commutes with every element of \( G \), this property extends to combining elements formed from the generator and center elements.
The primary understanding is that due to the regularity imposed by the cyclic nature and the central elements, every multiplication conforms to commutative laws. This sets \( G \) apart as abelian, as demonstrated through step-by-step verification of commutativity across any product of elements derived from \( a^n \) and a central element. Abelian groups offer simpler mathematical structures and are prevalent in various applications due to their ordered interaction.

Commutative Property

The commutative property refers to the principle where the order of operations does not alter the result. In the context of group theory, it implies that for any elements \( g \) and \( h \) in a group \( G \), the equation \( gh = hg \) always holds true.
This fundamental property is pivotal when discussing abelian groups. If a group is abelian, by the definition, it satisfies the commutative property.

• This property simplifies the group operation, offering predictability and allowing for vast simplifications in proofs and applications.
• In the problem at hand, the commutativity of elements within \( G \) was established due to the underlying structure imposed by the cyclic quotient group \( G/C \).

Without the commutative property, analyzing interactions and operations within groups would become infinitely more complex. However, with commutativity, many complex equations and problems reduce to straightforward calculations, preserving algebraic friendliness. This predicts an abelian-like simplicity in operations, aiding in understanding greater mathematical systems constructed around combinational group structures.