Problem 5 - A Book of Abstract Algebra

Question

Let \(a\) be any element of a group \(G\). Prove the following: If \(a\) has order \(n\), and \(a^{r}=a^{s}\), then \(n\) is a factor of \(r-s\)

Step-by-Step Solution

Answer

If \(a^r = a^s\), then \(n | (r-s)\), so \(n\) divides \(r-s\).

Step 1: Understand the Order of an Element

The order of an element \(a\) in a group \(G\) is the smallest positive integer \(n\) such that \(a^n = e\), where \(e\) is the identity element of \(G\). It means raising \(a\) to the power \(n\) returns the identity element.

Step 2: Use Given Equation

We are given that \(a^r = a^s\). To solve for the relationship between \(r\) and \(s\), we manipulate this equation. By multiplying both sides by \((a^s)^{-1}\), we obtain:\[a^r \cdot (a^s)^{-1} = e\].

Step 3: Simplify the Equation

Using the properties of exponents, \(a^r \cdot (a^s)^{-1} = a^{r-s}\). Thus, the equation from Step 2 becomes \(a^{r-s} = e\).

Step 4: Apply Definition of Order

Since \(a^{r-s} = e\) and \(a^n = e\) (with \(n\) as the minimal such integer), it follows from the definition of order that \(n\) divides \(r-s\). This is because \(r-s\) is another integer for which raising \(a\) to its power returns the identity, hence \(r-s\) must be a multiple of \(n\).

Key Concepts

Order of an Element

The order of an element in group theory is a fundamental concept that defines the "cyclic pattern" of a group element under the operation of the group. For an element \(a\) in a group \(G\), the order is the smallest positive integer \(n\) such that raising \(a\) to the exponent \(n\) results in the identity element of the group, notated as \(a^n = e\). This means:

• If \(a^n = e\), \(a\) cycles back to the identity after \(n\) operations.
• If no smaller positive integer achieves this, then \(n\) is indeed the order of \(a\).
• The identity element \(e\) acts as the 'reset' point, so reaching \(e\) confirms the completion of a full cycle.

This property helps in understanding the structure of the group as it provides insight into how elements interact through repetition.

Exponential Equations in Groups

In group theory, exponential equations are equations where elements are raised to exponents. These are essential for understanding relationships between group elements. Take the equation \(a^r = a^s\) in a group \(G\). This implies a relationship between the exponents \(r\) and \(s\). We solve this by:

• Recognizing both expressions represent the same group element.
• Rewriting the equation by canceling common terms through multiplication by inverses, like \(a^r \cdot (a^s)^{-1} = e\).
• Simplifying as \(a^{r-s} = e\), revealing that raising \(a\) to the power of \(r-s\) results in the identity.

This satisfies the order condition of the group element: \(r-s\) must be a multiple of the order \(n\), since raising \(a\) to a multiple of its order always results in the identity.

Identity Element in Groups

The identity element is a cornerstone of group structure. In any group \(G\), the identity element, denoted as \(e\), is the unique element that leaves other elements unchanged when combined with them. Formally, for any element \(a\) in \(G\), the following holds:

• \(a \cdot e = a\)
• \(e \cdot a = a\)

In the context of the original exercise, the identity plays a crucial role.

• The order of an element is determined by when the element, under repetition of the group operation, returns to the identity \(e\).
• When an exponent equation like \(a^{r-s} = e\) is established, it confirms that \(r-s\) forms a complete cycle, recognizable by returning to the identity.

This reinforces the pervasive role of the identity element in bridging group operations and the cyclic properties of its elements.