Question

Sei \(G\) eine abelsche Gruppe. (a) Wenn \(g\) und \(h\) zwei Elemente der Ordnung 2 in \(G\) sind, so hat auch \(g h\) die Ordnung \(2 .\) (b) Zeigen Sie: Zwei verschiedene Elemente der Ordnung 2 in \(G\) erzeugen eine Untergruppe der Ordnung 4 . (c) Zeigen Sie: Sei \(G\) eine zyklische Gruppe gerader Ordnung. Dann enthält \(G\) genau ein Element der Ordnung 2 .

Short answer

(a) The product \(gh\) also has order 2. (b) Two elements of order 2 form a subgroup of order 4. (c) In a cyclic group of even order, there is exactly one element of order 2.

Step-by-step solution

Understanding the Problem

We are required to prove statements about elements of order 2 in an abelian group and to consider subgroup properties. Specifically, we need to address three parts of the problem related to elements of order 2.

Understanding Elements of Order 2

An element has order 2 if when squared it equals the identity element, i.e., for an element \( g \), if \( g^2 = e \), where \( e \) is the identity of the group.

Proof of (a) If g and h have Order 2, then gh has Order 2

Since \( g^2 = e \) and \( h^2 = e \), we compute \((gh)^2 = ghgh = g(hg)h = g^2h^2\). In an abelian group, \( hg = gh \), hence \( (gh)^2 = g^2h^2 = ee = e \). So, \( gh \) has order 2.

Proof of (b) Elements of Order 2 Form a Subgroup of Order 4

Let \( g \) and \( h \) be two distinct elements of order 2. The subgroup generated by \( g \) and \( h \) is \( \{e, g, h, gh\} \). All elements are unique: \( g \), \( h \), and \( gh \) are distinct because \( g eq h \) and none equal the identity \( e \). Therefore, the subgroup has 4 elements.

Proof of (c) Cyclic Group of Even Order Has One Element of Order 2

Let \( G = \langle a \rangle \), where \( a \) generates \( G \) and \( |G| = 2n \). The element \( a^n \) has order 2, as \((a^n)^2 = a^{2n} = e \). Since the group is cyclic, \( G \) consists of powers of \( a \), and there can only be one element (namely \( a^n \)) whose square is the identity.

Key concepts

Elements of Order 2

In group theory, an element is said to have an order of 2 if, when multiplied by itself, it returns the identity element of the group. In mathematical notation, for an element \( g \) in a group \( G \), it means that \( g^2 = e \), where \( e \) is the identity element of \( G \). This means that applying the group operation to \( g \) twice gives you \( e \).

In an abelian group, where all elements commute with one another, interesting properties arise when working with elements of order 2. For example, if \( g \) and \( h \) are both elements of order 2 in an abelian group \( G \), then their product, \( gh \), will also have order 2. This follows from the fact that \( (gh)^2 = g^2h^2 = ee = e \), thanks to the commutative property \( gh = hg \).

This property flows naturally from the notion that if you swap the elements around in their multiplication, the product remains unaffected. When elements of the same order interact under these rules, it provides results that are consistent with their defined order.

Subgroups in Group Theory

A subgroup is a smaller group contained within a larger group that itself satisfies the group properties: closure, associativity, identity, and inverse elements. This means a subgroup is a subset of a group that can function independently as a group. When dealing with elements of order 2, particularly in abelian groups, interesting subgroups can be formed.

Consider two distinct elements \( g \) and \( h \), each of order 2. The subgroup formed by them, alongside the identity element, is \( \{e, g, h, gh\} \). This subgroup satisfies all the subgroup criteria because:

• The identity \( e \) is present.
• Each element is invertible by itself (since both \( g \) and \( h \) have order 2).
• The product of any two elements still lies within the group, like \( gh \), which is distinct and of order 2.

This subgroup, called a Klein four-group when it consists of four elements, exemplifies a typical formation seen with this element type.

Cyclic Groups

A cyclic group is one that can be generated by a single element, meaning every element of the group can be expressed as powers of this generator. Mathematically, if \( G \) is a cyclic group, then there exists an element \( a \) such that every element \( g \) in \( G \) can be written as \( a^k \).

When a cyclic group has an even order, say \( 2n \), it contains exactly one element of order 2. If \( a \) is the generator of \( G \), then the element \( a^n \) has an order of 2, because \( (a^n)^2 = a^{2n} = e \). This element is unique because a cyclic group is based on the powers of a single generator, and only \( a^n \), when multiplied by itself, will yield the identity element. Thus, in cyclic groups of even size, the presence of a single element of order 2 is a defining characteristic.