Problem 2 - A Book of Abstract Algebra

Question

A property of groups is said to be "preserved under homomorphism" if, whenever a group \(G\) has that property, every homomorphic image of \(G\) does also. In this exercise set, we will survey a few typical properties preserved under homomorphism. If \(f: G \rightarrow H\) is a homomorphism of \(G\) onto \(H\), prove each of the following: If \(G\) is cyclic, then \(H\) is cyclic.

Step-by-Step Solution

Answer

If \(G\) is cyclic and \(f: G \rightarrow H\) is a surjective homomorphism, then \(H\) is cyclic, generated by \(f(g)\).

Step 1: Understand the Concepts

A group \(G\) is cyclic if there exists an element \(g \in G\) such that every element of \(G\) can be written as \(g^n\) for some integer \(n\). A homomorphism \(f: G \rightarrow H\) is a function that respects the group operation, meaning \(f(g_1 \cdot g_2) = f(g_1) \cdot f(g_2)\) for any elements \(g_1, g_2 \in G\). To say \(f\) is surjective (onto) means every element \(h \in H\) has a preimage in \(G\). We aim to prove if \(G\) is cyclic, \(H\) is also cyclic under these conditions.

Step 2: Define the Cyclic Property of G

Assume \(G\) is cyclic, generated by element \(g\), i.e., \(G = \langle g \rangle\). This means every element of \(G\) can be expressed in the form \(g^n\) for some integer \(n\).

Step 3: Apply the Homomorphism

The homomorphism \(f: G \rightarrow H\) is surjective, so for every element \(h \in H\), there is an element \(x \in G\) such that \(f(x) = h\). Specifically, for each \(g^n \in G\), compute \(f(g^n) = f(g)^n\). We denote \(f(g) = h_0\), which is some element of \(H\).

Step 4: Show H is Generated by f(g)

Since \(f\) is onto, every element in \(H\) is of the form \(f(g^n) = (f(g))^n\) because \(f\) respects the group operation \((f(g^m) \cdot f(g^n) = f(g^{m+n}))\). Therefore, \(H\) is generated by the single element \(h_0 = f(g)\), indicating \(H\) is cyclic.

Key Concepts

Cyclic Groups

Cyclic groups are an essential concept in group theory and appear frequently in mathematical problems. A group is called cyclic if there exists a single element, known as a generator, which can produce every element of the group through repeated application of the group operation. In formal terms, a group \( G \) is cyclic if there is an element \( g \) in \( G \) such that every element \( x \) in \( G \) can be written as \( g^n \) for some integer \( n \).
This means if you choose a generator wisely, you can "build" the entire group from it.

• Generators play a crucial role in understanding the structure and order of cyclic groups.
• The simplest example of a cyclic group is the group of integers under addition, \( \mathbb{Z} \), which is generated by the element 1.
• Another notable example is the multiplicative group of units modulo \( n \), denoted as \( \mathbb{Z}_n^* \).

Group Homomorphism

A group homomorphism is a special type of function between two groups that respects the group structure. In simpler terms, if \( f: G \rightarrow H \) is a homomorphism from group \( G \) to group \( H \), this means for any two elements \( a, b \in G \), the function satisfies the property \( f(a \cdot b) = f(a) \cdot f(b) \).
This property ensures that the group operation is maintained under the function.

• Homomorphisms play a key role in linking different groups and analyzing their properties.
• An important aspect of homomorphisms is whether they are injective (one-to-one) or surjective (onto).
• Surjective homomorphisms allow every element in the target group \( H \) to be possibly mapped from the source group \( G \).

Homomorphic Image

The concept of a homomorphic image comes into play when looking at the result of applying a group homomorphism to a group.
The homomorphic image of a group \( G \) through a homomorphism \( f: G \rightarrow H \) is the subgroup \( H \) itself, assuming \( f \) is surjective.

• This image retains much of the original group's structure, but it may lose certain details depending on \( f \).
• It is the output of the group \( G \) processed through the function \( f \).
• The image group \( H \) inherits any properties of \( G \) that are preserved under homomorphism, like the cyclic property.

Cyclic Property

The cyclic property, in the context of group theory, specifies that a group's structure can be generated by a single element.
When a group is cyclic, any homomorphic image of the group will also exhibit this cyclic nature, given certain conditions are met, such as surjectivity.

• If \( G \) is cyclic and \( f: G \rightarrow H \) is a surjective homomorphism, then \( H \) will be cyclic as well.
• This holds because if \( G \) is generated by \( g \), then \( f(g) \) will generate \( H \).
• The cyclic nature is a perfect example of a property preserved under homomorphism, showing the power and boundaries of mapping between groups.