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Groups

While the integers \(\mathbb Z\), the real numbers \(\mathbb R\) or the complex numbers \(\mathbb C\) are examples of algebraic structures known as number systems, there are very many important algebraic structures that are not number systems, but have similar properties. One of them are groups, to which this chapter is dedicated.

Surprisingly, groups often appear, whenever an operation on a set creates another element of this set. For instance, the operation addition, together with any of the number systems \(\mathbb Z\), \(\mathbb R\) and \(\mathbb C\), generates the additive groups \((\mathbb Z, + )\), \((\mathbb R, + )\) and \((\mathbb C, + )\). This is because an addition (even a repeated one) of two numbers of these number systems generates another number of the same number system (e.g. the addition of two integers \(x,y\in\mathbb Z\) will always generate an integer, but never a fraction). Together with the operation “addition”, all these number systems also share the property that they contain one designated number, called zero, which does not change any number of this system, when it is added to it. On the other hand, for each number there exists its inverse (called its negative number), which changes it into zero, when it is added to it. Actually, this is an all-pervasive connection of negative numbers and the number zero, which is common in all number systems and also in all other groups.

In the general sense, we might think of some operations themselves as “objects” of a specific set and investigate, in which cases the cascading application of two such operations is “identical” to another operation in this set.

As an example, consider the set \(S\) of six symmetries of a equilateral triangle, shown in the following figure:

Let us now interpret the operation “\( + \)” as the *composition of these symmetries” (note that “\( + \)” now does not mean the usual addition known from the high school but another operation). With this new kind of operation, we have established the group \((S, + )\). It now behaves like a number system. This is because the composition “\( + \)” (even a repeated one) of any two symmetries of \(S\) generates another symmetry in \(S\). To see it, take the reflection \(e\), followed by the reflection \(d\), which will results in the same symmetry as the rotation \(b\) is: \[b=d+e.\]
To see it, let us denote the edges of the original triangle by \( A B C \), and apply \(b\) on such a denoted triangle, and comparing the resulting triangle with the composed application of \(e\) and \(d\). As we can see, both operations change the denoted triangle in the same way: \[b=b(A B C) , \text{resulting in the triangle } C A B\] \[d + e=d + e(A B C)=d(C B A), \text{resulting in the same triangle } C A B.\]

Further note that the symmetry \(a\) behaves just like it was (actually it is) the zero number of the group (i.e. nothing happens to the resulting triangle, if we compose \(a\) to any of the symmetries), e.g.
\[a+e=a+e(A B C)=a(C B A), \text{leaving the triangle } C B A \text{ unchanged}.\].

On the other hand, all reflections have also “negative” reflections, just like they were numbers. For instance, the inverse reflections of \(d,e,f\) are the same reflections, i.e. composing one such reflection twice will result in the unchanged triangle (i.e. the symmetry \(a\)):
\[d + d=a,\] \[e + e=a,\] \[f + f=a,\]
On the other hand, the two rotations \(b,c\) are negative to each other:
\[b + c=a\] \[c +b =a.\]

This demonstrates that one can “calculate” with elements of general groups as one is used to in number systems. In this sense, groups are more general algebraic structures than number systems. Moreover, wherever structure-preserving symmetries appear, groups follow close behind.

| | | | Contributors: bookofproofs

1.Symmetry Groups

2.Axiom: Axiom of Existence of Inverse Elements

3.Axiom: Axiom of Commutativity

4.Definition: Subgroup

5.Definition: Group

6.Theorem: Construction of Groups from Commutative and Cancellative Semigroups


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