Question

If \(\mathcal{A}\) is a group in which every element other than the identity, \(I\), has order 2, prove that \(\mathcal{A}\) is Abelian. Hence show that if \(X\) and \(Y\) are distinct elements of \(\mathcal{A}\), neither being equal to the identity, then the set \(\\{I, X, Y, X Y\\}\) forms a subgroup of \(\mathcal{A}\). Deduce that if \(\mathcal{B}\) is a group of order \(2 p\), with \(p\) a prime greater than 2, then \(\mathcal{B}\) must contain an element of order \(p .\)

Short answer

Group \(\bigmathcal{A}\) is Abelian since \(ab = ba\). The set \{I, X, Y, XY\}\ is a subgroup of \(\bigmathcal{A}\). Group \(\bigmathcal{B}\) contains an element of order \ p\.

Step-by-step solution

- Element Properties in Group \(\bigmathcal{A}\)

Identify the properties of elements in group \(\bigmathcal{A}\). Given that every element other than the identity has order 2, it means for any element \(a eq I\), \(a^2 = I\).

- Commutativity Check for Abelian

To prove group \(\bigmathcal{A}\) is Abelian, show that for any two elements \(a, b eq I\), \(ab = ba\). Consider the product \(ab eq I\). Since \(ab \cdot ab = I,\), then \(ab = (ab)^{-1} = b^{-1}a^{-1} = ba\), proving \(\bigmathcal{A}\) is Abelian.

- Subgroup Formation

Given distinct elements \(X, Y eq I\) in \(\bigmathcal{A}\), show that the set \{I, X, Y, XY\}\ forms a subgroup. Verify closure, identity, inverses, and associativity. Closure: \(XY eq I\) as \(X, Y\) are distinct and have order 2. Identity \(I\) is included. Inverses: \(I^{-1} = I, X^{-1} = X, Y^{-1} = Y, (XY)^{-1} = Y^{-1}X^{-1} = YX = XY\). Associativity follows from \(\bigmathcal{A}\). Thus, \{I, X, Y, XY\}\ forms a subgroup.

- Group of Order \2p\

Given \(\bigmathcal{B}\) is a group of order \2p\ with prime \ p > 2\, show \(\bigmathcal{B}\) contains an element of order \ p\. Apply Cauchy's theorem stating that for a prime \ p \ dividing the order of a finite group, there exists an element of that order. Since \ p \ divides \ 2p\, an element of order \ p\ exists in \(\bigmathcal{B}\).

Key concepts

Abelian Groups

In group theory, an Abelian group is one where the group operation is commutative. This means that for any two elements \(a\) and \(b\) in the group, the equation \(ab = ba\) holds true. For example, consider the integers under addition. Regardless of the order in which you add two integers, the result is the same. In the exercise, we identified that if each element (other than the identity) has order 2, it implies that each element is its own inverse. When we tested the product \(ab\) in this group, we found it satisfies commutativity, hence confirming the group is Abelian.

Orders of Elements

The order of an element in a group is the smallest positive integer \(n\) such that \(a^n = I\), where \(I\) is the identity element and \(a\) is a group element. In our problem, we have a group \(\bigmathcal{A}\) where each element other than the identity has order 2. This means for any element \(a\) in \(\bigmathcal{A}\), we have \(a^2 = I\). Such elements are their own inverses, simplifying the structure of the group. This specific type of order (2) played a key role in proving the group is Abelian and helped us verify the properties required to form subgroups.

Subgroups

A subgroup is a subset of a group that is itself a group under the same operation. To establish that a set forms a subgroup, one must verify four criteria: closure, identity, inverses, and associativity. In the exercise, we considered the set \(\{I, X, Y, XY\}\), where \(X\) and \(Y\) are distinct elements of a group \(\bigmathcal{A}\) with order 2 elements. We showed:

• Closure: The product \(XY\) remains in the set.
• Identity: The identity element \(I\) is included.
• Inverses: All elements are their own inverses.
• Associativity: Inherited from the group \(\bigmathcal{A}\).

Thus, \(\{I, X, Y, XY\}\) forms a valid subgroup.

Cauchy's Theorem

Cauchy's theorem is a fundamental result in group theory. It states that if a prime \(p\) divides the order of a finite group \(\bigmathcal{B}\), then there exists an element in \(\bigmathcal{B}\) of order \(p\). For example, in a group of order \(2p\) with \(p\) a prime greater than 2, since \(p\) divides \(2p\), Cauchy's theorem guarantees the existence of an element in the group with order \(p\). This principle was used in the exercise to conclude that a group of such order must contain an element of order \(p\), reinforcing the broad applicability of the theorem in various contexts within group theory.