Problem 1 - A Book of Abstract Algebra

Question

Let \(G\) be an abelian group. If \(H=\left\\{x \in G: x=x^{-1}\right\\}\), that is, \(H\) consists of all the elements of \(G\) which are their own inverses, prove that \(H\) is a subgroup of \(G\).

Step-by-Step Solution

Answer

The set \(H\) is a subgroup of \(G\) because it is closed under the group operation, contains the identity element, and is closed under inverses.

Step 1: Definition of a Subgroup

To prove that a subset like \(H\) is a subgroup of \(G\), we need to show three things: 1) \(H\) contains the identity element of \(G\). 2) \(H\) is closed under the group operation. 3) \(H\) is closed under taking inverses.

Step 2: Check for the Identity Element

In any group, the identity element \(e\) satisfies \(e = e^{-1}\). Thus, since \(e\) is its own inverse, \(e \in H\). This means that the identity element of \(G\) is in \(H\).

Step 3: Closure Under Group Operation

We need to show that if \(a, b \in H\), then their product \(ab \in H\). Since \(a = a^{-1}\) and \(b = b^{-1}\), we have \((ab)^{-1} = b^{-1}a^{-1} = ba\). But since \(G\) is abelian, \(ab = ba\), so \((ab) = (ab)^{-1}\). Hence, \(ab \in H\).

Step 4: Closure Under Inverses

We need to show that for any element \(a \in H\), its inverse \(a^{-1} \in H\). By definition of \(H\), \(a = a^{-1}\). Hence, \(a^{-1} = a \in H\). This satisfies the condition for closure under taking inverses.

Step 5: Conclusion

Having verified all the conditions (identity in \(H\), closure under group operation, and closure under inverses), we can conclude that \(H\) is a subgroup of \(G\).

Key Concepts

Abelian Group

An Abelian group, also known as a commutative group, is a fundamental concept in group theory. In an Abelian group, the order in which you perform the group operation doesn't matter. For any two elements, say \(a\) and \(b\), within the group, the equation \(a \cdot b = b \cdot a\) always holds. This property brings about a sense of symmetry and simplifies many group computations.

If you imagine a set of numbers with a group operation like addition, then in Abelian groups every pair of numbers can be added in any order without affecting the sum. This definition extends beyond numbers to more abstract objects in mathematics.

Abelian groups are everywhere in mathematics and they show up in various contexts, whether that's dealing with integers, polynomials, or more abstract algebraic structures. Understanding them helps set a foundation for exploring deeper topics in group theory.

Subgroup

A subgroup is a smaller group within a larger group that itself satisfies the conditions required to be a group. To determine if a subset \(H\) of a group \(G\) forms a subgroup, it must also be a group under the same operation as \(G\). This subgroup will inherit the group's main operation and identity element.

There are three main criteria to verify if \(H\) is a subgroup:

• Contains the Identity Element: \(H\) must include the identity element of \(G\).
• Closure Under Group Operation: For any elements \(a\) and \(b\) in \(H\), \(a \cdot b\) must also be in \(H\).
• Closed Under Inverses: For any element \(a\) in \(H\), \(a^{-1}\) must also be in \(H\).

By showing these properties hold for a subset, you prove it's a subgroup, preserving the structure and operation of the original group.

Group Operation

A group operation is a way of combining two elements from a set to produce another element from the same set. This operation, usually denoted by \(\cdot\), could be anything from addition or multiplication to more complex mathematical functions.

For an operation to qualify as a group operation, it must satisfy four fundamental properties across all elements in the group:

• Closure: Performing the operation on any two elements of the group results in another element within the group.
• Associativity: The grouping of elements doesn't affect the result of the operation. For any elements \(a, b, c\), \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Identity Element: There is an element in the group, let's call it \(e\), such that for any element \(a\), \(e \cdot a = a = a \cdot e\).
• Inverses: For every element \(a\), there is an element \(a^{-1}\) such that \(a \cdot a^{-1} = e\).

This operation is the backbone of the group's structure, determining how the elements interact and ensuring the group properties are maintained.

Identity Element

In group theory, the identity element is the element that does nothing when combined with any other element in the group. Think of it as the 'neutral' element in the group operation. It's the number zero for addition or one for multiplication.

The identity element, often denoted by \(e\) or \(0\) or \(1\), has a distinctive property: For any element \(a\) in the group, the operation of \(a\) with \(e\) (i.e., \(a \cdot e \)) results in \(a\) itself. Similarly, \(e \cdot a = a\), showcasing the element's neutral role in both positions.

This element is crucial when considering the group's closure under their operation and is always included when checking subgroup properties. Every single element has its inverse, which, when applied in the operation with the element, will result in the identity element, maintaining group structure.