Problem 1 - A Book of Abstract Algebra

Question

Let \(G\) be a group and let \(a, b \in G\). Prove the following: If \(a\) is a power of \(b\), say \(a=b^{k}\), then \(\langle a\rangle \subseteq\langle b\rangle\).

Step-by-Step Solution

Answer

Since \( a = b^k \), any element from \( \langle a \rangle \) is of the form \( b^{mk} \) and belongs to \( \langle b \rangle \). Thus, \( \langle a \rangle \subseteq \langle b \rangle \).

Step 1: Understand Subgroup Generation

A subgroup generated by an element, say \( \langle a \rangle \), is the set of all integer powers of \( a \). For \( \langle a \rangle \), this means \( \{ a^n ext{ such that } n \in \mathbb{Z} \} \). We want to show every element of this set is also in \( \langle b \rangle \).

Step 2: Expressing Elements in Terms of b

Given that \( a = b^k \), substitute \( a \) in any power format. If \( x \in \langle a \rangle \), then \( x = a^m \) for some integer \( m \). Therefore, substitute for \( a \), \( x = (b^k)^m \).

Step 3: Simplify Expression for x

Simplify \( x = (b^k)^m \) using the exponentiation rule \((b^k)^m = b^{km}\). This shows that any power of \( a \), \( a^m \), can be rewritten as a power of \( b \), specifically \( b^{km} \).

Step 4: Establish Inclusion in Subgroup

Since every element \( x = a^m \) in \( \langle a \rangle \) can be represented as \( x = b^{km} \), and \( b^{km} \) is obviously an element of \( \langle b \rangle \) because it is a power of \( b \), we conclude \( x \in \langle b \rangle \). Therefore, every element in \( \langle a \rangle \) is also in \( \langle b \rangle \).

Key Concepts

Subgroup

In the realm of group theory, a **subgroup** is essentially a smaller group contained within a larger group. When we talk about a subgroup, we're talking about a set of elements that itself forms a group according to the same operation as the larger group.
To be a subgroup, certain conditions must be met:

• The identity element of the larger group must be in the subgroup.
• If you take any elements in the subgroup and combine them (using the group operation), the result must also be in the subgroup.
• For every element in the subgroup, its inverse must also be a part of the subgroup.

Understanding these properties is crucial when dealing with concepts like subgroup generation and powers of elements.

Exponentiation

**Exponentiation** in group theory doesn't differ much in spirit from regular exponentiation you know from arithmetic. It's a way to apply the group operation repeatedly.
For a group element \( g \), if you exponentiate it to the integer \( n \), you perform the group operation \( n \) times. Mathematically, this means \( g^n \) is defined as \( g \cdot g \cdot \ldots \cdot g \) (\( n \) times). For negative exponents, it involves the inverse: \( g^{-n} = (g^{-1})^n \).
This operation is powerful as it helps to generate subgroups and explore properties like closure, associates, and identities within the context of group structure.

Subgroup Generation

**Subgroup generation** is a method to construct a subgroup from one or more elements.
When we say "a subgroup is generated by element \( a \)," we denote this as \( \langle a \rangle \). This set comprises all possible powers of \( a \), including positive, negative, and zero powers.
To generate a subgroup:

• Start with an element, say \( a \).
• Compute all integer powers of \( a \) using the group operation, such as \( a^1, a^2, a^{-1} \) etc.
• The collection of these powers forms \( \langle a \rangle \), which is a subgroup of the original group \( G \).

This process not only illustrates the concept of a subgroup but also connects directly to understanding the powers of elements, providing an insight into the structure and potential symmetries within the group.

Powers of Elements

The **powers of elements** in a group are defined in terms of repeated application of the group operation. For any element \( a \) in a group \( G \), we denote its powers as \( a^n \) where \( n \) is an integer.
For example:

• \( a^0 \) is the identity element of the group, regardless of \( a \).
• \( a^1 \) is the element itself, \( a \).
• \( a^2 \) means combining \( a \) with itself once more using the group operation.

The interesting feature of the powers of elements lies in their ability to demonstrate group structure. For instance, if \( a=b^k \), each power of \( a \) is also a power of \( b \). Consequently, this implies that the subgroup generated by \( a \) is contained within the subgroup generated by \( b \). Such relationships are vital for understanding the hierarchy and properties of subgroups in a larger group.