Problem 4 - A Book of Abstract Algebra

Question

If \(x a y=a^{-1}\), then \(y a x=a^{-1}\)

Step-by-Step Solution

Answer

Given the operation is commutative, \(y a x = x a y = a^{-1}\).

Step 1: Understand the operation

Given the operation \(x a y = a^{-1}\), it implies combining \(x\) and \(y\) with operation \(a\) results in \(a^{-1}\). We are asked to show that reversing the order, \(y a x\), also results in \(a^{-1}\).

Step 2: Recognize properties of operations

Operations such as multiplication have the commutative property, meaning \(x \cdot y = y \cdot x\). For operation \(a\) to have \(x a y = a^{-1}\), \(y a x\) must yield the same result if \(a\) is also commutative.

Step 3: Assume the operation is commutative

Assuming \(a\) is a commutative operation, we have \(x a y = y a x\). Thus, \(x a y = a^{-1}\) implies \(y a x = a^{-1}\) because the order of elements doesn't affect the result.

Step 4: Conclusion

Under the assumption that operation \(a\) is commutative, swapping \(x\) and \(y\) doesn't change the result. Therefore, \(y a x = a^{-1}\) as well.

Key Concepts

Commutative Property

The commutative property is a fundamental concept in abstract algebra and mathematics as a whole. It refers to the ability of certain operations, like addition and multiplication, to yield the same result regardless of the order of the operands. This means that if an operation is commutative, changing the order of the elements does not affect the outcome. For example:

• Addition: For any two numbers, say 3 and 5, it holds that \(3 + 5 = 5 + 3\).
• Multiplication: Similarly, for any two numbers, \(7 \times 2 = 2 \times 7\).

In the context of the original exercise, the operation \(a\) is assumed to be commutative. This assumption allows us to state that swapping the elements in the operation \(x a y\) to become \(y a x\) does not change the result. Therefore, if \(x a y = a^{-1}\), then \(y a x = a^{-1}\) as well. This is only possible because of the inherent nature of commutative operations, which apply to numbers, algebraic structures, and more.

Inverse Elements

Inverse elements are crucial in understanding group theory, a core part of abstract algebra. An inverse element essentially "undoes" the effect of another element, paired under a specific operation. For a given element \(a\), its inverse under a binary operation is typically denoted \(a^{-1}\). Together, these two elements yield an identity element (like zero for addition or one for multiplication) when combined with their operation. For example:

• Additive Inverse: For the number 5, its additive inverse is -5, since \(5 + (-5) = 0\).
• Multiplicative Inverse: For the number 7, its multiplicative inverse is \(\frac{1}{7}\), since \(7 \times \frac{1}{7} = 1\).

In the problem statement, \(a^{-1}\) is utilized as the result of the operation when \(x\) and \(y\) are combined with \(a\). It tells us that there exists a specific element that, when used with \(a\), results in a particular identity element designated as \(a^{-1}\). Understanding inverse elements helps in comprehending how various mathematical elements can be manipulated to achieve a specific outcome.

Binary Operations

A binary operation is a key concept in algebra, referring to an operation that combines two elements (say \(x\) and \(y\)) to produce another element within the same set. Some commonly known binary operations include addition, subtraction, multiplication, and division.

To be considered a binary operation, the operation itself must satisfy specific properties, and its result must remain within the set from which the elements were chosen. For instance:

• Addition and multiplication with real numbers are binary operations because combining any two real numbers results in another real number.
• Subtraction of integers also stays within the integer set, hence, it is a valid binary operation.

In the exercise, the operation denoted as \(a\) must be a binary operation since it combines two elements, \(x\) and \(y\), resulting in another element, \(a^{-1}\). It is this characteristic as a binary operation that allows us to manipulate and understand the operation using established properties like the commutative property. This understanding forms the basis for exploring more complex algebraic structures and operations.