Group/Linear algebra/Introduction/Section

In linear algebra, we work in general over a fixed base field ${}K$. The most important field for us is the field of real numbers ${}\mathbb {R}$, which we have already used and which is introduced in analysis in an axiomatic way. A field is characterized by the existence of two binary operations fulfilling certain properties, namely addition and multiplication. Both these operations (for multiplication, one has to remove ${}0$) are instances of an important algebraic structure: a group.

Definition

A set ${}G$ together with a special element ${}e\in G$ and with a binary operation

$G\times G\longrightarrow G,(g,h)\longmapsto g\circ h$

is called a group when the following properties are fulfilled.

1. The binary operation is associative, i.e., for all ${}f,g,h\in G$, we have

   ${}(f\circ g)\circ h=f\circ (g\circ h)\,.$

2. The element ${}e$ is a neutral element, i.e., for all ${}g\in G$, we have

   ${}g\circ e=g=e\circ g\,.$

3. For every ${}g\in G$, there exists an inverse element, i.e., there exists some ${}h\in G$ such that

   ${}h\circ g=g\circ h=e\,.$

A group is called commutative if the operation is commutative. Important examples for commutative groups are ${}(\mathbb {Z} ,0,+)$, ${}(\mathbb {R} ,0,+)$, ${}(\mathbb {R} \setminus \{0\},1,\cdot )$ or ${}(\mathbb {R} ^{n},0,+)$ with the componentwise zero

${}0=(0,0,\ldots ,0)\,$

and componentwise addition.

In a group ${}(G,e,\circ )$, the neutral element is uniquely determined. For if ${}e'$ is another element fulfilling the characteristic property of the neutral element, meaning

${}x\circ e'=e'\circ x=x\,$

for all ${}x\in G$, then we can directly deduce

${}e=e\circ e'=e'\,.$

Lemma

Let ${}(G,e,\circ )$ be a group. Then, for every ${}x\in G$, the element ${}y\in G$ fulfilling

${}x\circ y=y\circ x=e\,$

is uniquely determined.

Proof

Let

${}x\circ y=y\circ x=e\,$

and

${}x\circ z=z\circ x=e\,.$

Then we have

${}y=y\circ e=y\circ (x\circ z)=(y\circ x)\circ z=e\circ z=z\,.$

$\Box$

Abstract structures like a set, a mapping, a binary operation, or a group have a double life. On one hand, they are really just the given formal structure; the elements are just some elements in a somehow given set, a binary operation is just any binary operation, and one should not imagine anything concrete. The symbols chosen are arbitrary and without any meaning. On the other hand, these abstract structures gain a second life in that many concrete mathematical structures obey the abstract properties. These concrete structures are examples or models for the abstract structure (and they are also a motivation to introduce the abstract structure). Both viewpoints are important, and one should always try not to confuse them.

Group theory is a branch of mathematics on its own, which we will not develop here systematically. Instead, we work with rings, and in particular with fields.