Dropping the commutativity of the operation makes groups less easy to work with, but allows to cover the common properties of such an important class of structures as sets of bijections (of various kinds) of a set to itself.

A

groupis a pair where is a set and a binary operation on that is

- associative, i.e. for all

and thus we can drop brackets without introducing any ambiguity;- there exists a neutral element, i.e. a special element satisfying for all It must not be confused with the integer number 1, as in general it has nothing to do with although it shares similar properties (rationals w.r.t. addition are an example of a group, and there these two things are indeed the same).
- for any there exists such that We denote by

To explain the title, note that these axioms are essentially the axioms of an abelian group, with the commutativity axiom dropped, and the operation written as instead of (Respectively, the notaton for the neutral element and for the inverse is adjusted.) We will often simply drop the from expressions, i.e. write instead of etc.

A number of useful rules can be easily derived from the axioms, e.g. the following:

The equation for any given has unique solution

Thus we can cancel: iff , for all

Indeed, by associativity and properties of the inverse and the neutral element, we have thus such an exists and is unique.

This immediately implies e.g. the following:

- is unique, for any given
- is the only neutral element in

The next important concept is of **homomorphism** and **isomorphism** of groups. These should come as no surprise after the abelian group homomorphisms and isomorphisms, and ring homomorphisms and isomorphisms.

Given two groups and , a function satisfying is called a

homomorphismfrom toWhen is a bijection, then it is called

isomorphism, and one writes , or just

From the point of view of abstract algebra, two isomorphic groups are hardly distinguishable, although of course they can appear in different disguises.

**Groups of bijections**

Quite often, nonabelian groups arise in the following context. Let be a set and be a set of bijections satisfying the following properties:

- if then the composition of and is in
- the identity bijection, is in
- if then the inverse of is in

Then is a group, usually denoted to emphasise the role of When we call a group of *permutations* (as bijections of a finite set are just permutations.)

This gives **examples of nonabelian groups**

- the group of all bijections (a.k.a. permutations) of the finite set usually called the symmetric group (of -element set). Note that is abelian, of order but ceases to be abelian for More generally,
- Let be an -dimensional vector space over a field The linear bijections form a group, denoted by or and called the general linear group over (in dimension )
- Dihedral group : the group of symmetries of the regular gon. It consists of elements, namely cyclic rotations and reflections w.r.t. symmetry axes. Sometimes also denoted by The group is isomorphic to , and the group is isomorphic to

Now let us show that

everygroup is isomorphic to a group of bijections.

Observe that for any the mapping defined by for any is a bijection (called * multiplication by on the left*). Moreover, for one has as by associativity of the multiplication Similarly, as Thus satisfies the group axioms, and is a group of bijections. The mapping is a group isomorphism, as it is a bijection (check this!) and as as shown above.

**Example: and ** Let where denotes the permutation sending to for Then e.g.

New groups from old: **direct products** (just as in the case of rings and abelian groups):

Given two groups and define the group

, thedirect productof and where the operation is component-wise:

Similarly, one can define direct products of an arbitrary number of groups.

**Subgroups**

A

subgroupof is a subset of that is closed under the multiplication in (Notation: or , the latter does not exlcude ) This means that for any also , and immediately implies

Examples of subgroups:

- Let and Let Then it is called the
*stabiliser*of in

Note that the stabiliser of in is isomorphic to where - Let and Then is denoted by and is called the
*special linear group*. - Some subgroups of the direct product let Then

**Generating sets, etc.**

First, a bit of terminology: for a group and any , the power is well-defined, as follows:

- if then
- if then
- if then the fold product of

It can perfectly happen that while neither nor , e.g. for any

Let Define the subgroup generated by to be i.e. the intersection of all the subgroups of containing

One can show that consists ofwordsin i.e. finite products of the form

where and we denoted

**Cyclic subgroups.** These are subgroups generated by just one element, i.e. one should take and denote We recall (just the notation changes from the additive to the multiplicative) that

More generally,

A group is called

finitely generatedif there exists such that and

E.g. any finite group, e.g. is finitely generated. E.g. is not finitely generated.