91Ó°ÊÓ

In group theory, one of the foundational properties that needs to be verified for any mathematical structure to be considered a group is the closure property. This means that if you take any two elements from the set and perform the defined operation on them, the result should also be an element of the same set.
For the operation \( (a, b) * (c, d) = (ad + bc, bd) \), we need to ensure the product produces another element from the same set. Our set, \( \mathbb{R} \times \mathbb{R} \), excludes pairs where the second component is zero, so we must verify \( bd eq 0 \).
This requirement is fulfilled because the inputs \(b\) and \(d\) are non-zero, and the multiplication of two non-zero numbers is non-zero. Hence, this operation keeps us within the bounds of our set, confirming that the closure property is satisfied.

The identity element in a group is a special element that, when combined with any element of the set using the group operation, leaves the original element unchanged. For the operation given by \( (a, b) * (c, d) \), our task is to find an element \((e_1, e_2)\) such that:

• \( (a, b) * (e_1, e_2) = (a, b) \)

Solving this corresponds to solving the equations:

• \( ae_2 + be_1 = a \)
• \( be_2 = b \)

The solution gives \( e_1 = 0 \) and \( e_2 = 1 \), hence \((0, 1)\) acts as our identity element. In simpler terms, multiplying any element by \((0, 1)\) using our operation returns the start value unchanged, which confirms the presence of an identity element in our set.

For a set to form a group, every element must have an inverse. An inverse of an element \((a, b)\) in the set is another element \((c, d)\) such that when the operation \((a, b) * (c, d)\) is performed, the identity element \((0, 1)\) is the result. In other words:

• \( (a, b) * (c, d) = (0, 1) \)

Solving gives us:

• \( ad + bc = 0 \)
• \( bd = 1 \)

From \( bd = 1 \), we find \( d = \frac{1}{b} \), and substituting into \( ad + bc = 0 \), we have \( c = -\frac{a}{b} \).
This confirms every element \((a, b)\) has an inverse \((-\frac{a}{b}, \frac{1}{b})\) within our set, fulfilling the inverse element condition needed for group structure.

Associativity is a critical property that ensures the sequence in which operations are performed does not affect the result. For this property, consider the equation \((a, b) * ((c, d) * (e, f)) = ((a, b) * (c, d)) * (e, f)\). Let's break it down:
First, compute \((c, d) * (e, f)\):

• \((ce + df, df)\)

And now perform \((a, b) * (ce + df, df)\), resulting in:

• \((a(ce+df) + b(df), bdf)\)

Compare with the result when calculating \(((a, b) * (c, d)) * (e, f)\):

• \( ((ad + bc)e + (bd)f, (bd)f)\)

Upon simplifying, both sides of the equation match, demonstrating that our operation is associative for any elements in our specified set, thus verifying associativity.

Groups can be categorized into two types: abelian and non-abelian. An abelian group is one where the operation is commutative, meaning \((a, b) * (c, d) = (c, d) * (a, b)\) holds true for any pair of elements. For our defined operation: \((a, b) * (c, d) = (ad + bc, bd)\) and \((c, d) * (a, b) = (cd + ba, bd)\).
These equations show a notable difference: \(ad + bc\) is not equal to \(cd + ba\) generally, meaning the order of operation affects the result. Therefore, the set under our defined operation is not commutative, classifying it as a non-abelian group.
This non-commutative characteristic makes non-abelian groups distinct and is an important consideration in the analysis of more complex algebraic structures.