Nonabelian groups

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 group is a pair (G,\cdot), where G is a set and a binary operation \cdot on G that is

  • associative, i.e. x\cdot (y\cdot z)=(x\cdot y)\cdot z for all x,y,z\in G,
    and thus we can drop brackets x\cdot (y\cdot z)=x\cdot y\cdot z without introducing any ambiguity;
  • there exists 1=1_G\in G, a neutral element, i.e. a special element satisfying 1\cdot x=x\cdot 1=x for all x\in G. It must not be confused with the integer number 1, as in general it has nothing to do with 1_G, 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 x\in G there exists x'\in G such that x\cdot x'=x'\cdot x=1. We denote x' by x^{-1}.

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 \cdot instead of +. (Respectively, the notaton for the neutral element and for the inverse is adjusted.) We will often simply drop the \cdot from expressions, i.e. write ab instead of a\cdot b, etc.

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

The equation a\cdot x=b for any given a,b\in G has unique solution x=b\cdot a^{-1}.
Thus we can cancel: x\cdot y=x\cdot z iff y=z, for all x,y,z\in G.

Indeed, by associativity and properties of the inverse and the neutral element, we have a^{-1}(ax)=(a^{-1}a)x=1\cdot x=x=a^{-1}b, thus such an x exists and is unique.

This immediately implies e.g. the following:

  1. x^{-1} is unique, for any given x\in G
  2. 1\in G is the only neutral element in G

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 (A,\cdot _A) and (B,\cdot _B), a function \phi:B\to A satisfying \phi(b\cdot _B b')=\phi(b)\cdot _A\phi(b') is called a homomorphism from B to A.

When \phi:B\to A is a bijection, then it is called isomorphism, and one writes (A,\cdot _A)\cong (B,\cdot _B), or just A\cong B.

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 \Omega be a set and G be a set of bijections f:\Omega\to\Omega satisfying the following properties:

  • if f,g\in G then f\circ g, the composition of f and g, is in G
  • id, the identity bijection, is in G
  • if f\in G then f^{-1}, the inverse of f, is in G

Then G is a group, usually denoted (G,\Omega) to emphasise the role of \Omega. When |\Omega|<\infty, we call (G,\Omega) a group of permutations (as bijections of a finite set are just permutations.)

This gives examples of nonabelian groups

  1. S_n, the group of all bijections (a.k.a. permutations) of the finite set \{1,\dots,n\}, usually called the symmetric group (of n-element set). Note that S_2 is abelian, of order |S_2|=2, but ceases to be abelian for n\geq 3. More generally, |S_n|=n!
  2. Let V be an n-dimensional vector space over a field \mathbb{F}. The linear bijections V\to V form a group, denoted by GL_n(\mathbb{F}) or GL(n,\mathbb{F}), and called the general linear group over \mathbb{F} (in dimension n)
  3. Dihedral group D_{2n}: the group of symmetries of the regular n-gon. It consists of 2n elements, namely n cyclic rotations and n reflections w.r.t. n symmetry axes. Sometimes also denoted by D_n. The group D_6 is isomorphic to S_3, and the group D_4 is isomorphic to \mathbb{Z}_2\times \mathbb{Z}_2.

Now let us show that

every group G is isomorphic to a group of bijections.

Observe that for any g\in G the mapping L_g:G\to G defined by L_g(x)=gx, for any x\in G, is a bijection (called multiplication by g on the left). Moreover, for h\in G one has L_g\circ\L_h=L_{gh}, as by associativity of the multiplication L_g(L_h(x))=L_g(hx)=ghx=(gh)x=L_{gh}(x). Similarly, (L_g)^{-1}=L_{g^{-1}}, as L_g(L_{g^{-1}}(x))=L_g(g^{-1}x)=gg^{-1}x=x. Thus L_G:=\{L_g\mid g\in G\} satisfies the group axioms, and (L_G,G) is a group of bijections. The mapping \phi : G\to L_G is a group isomorphism, as it is a bijection (check this!) and as \phi(gh)=L_{gh}=L_g\circ L_h, as shown above.

Example: S_3 and L_{S_3}. Let G=S_3=\left\{ \begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}, \begin{pmatrix}1&2&3\\2&1&3\end{pmatrix}, \begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}\right., \left.\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}, \begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}, \begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}\right\}=\{\pi_1,\dots,\pi_6\}, where \begin{pmatrix}1&\dots&k\\ g_1&\dots& g_k\end{pmatrix} denotes the permutation sending \ell to g_\ell for 1\leq \ell\leq k. Then e.g. L_{\pi_2}=\begin{pmatrix}\pi_1&\pi_2&\pi_3&\pi_4&\pi_5&\pi_6\\ \pi_2&\pi_1&\pi_5&\pi_6&\pi_3&\pi_4\end{pmatrix}.

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

Given two groups (A,\cdot _A) and (B,\cdot _B), define the group
A\times B=(A\times B,\cdot ), the direct product of A and B, where the operation \cdot is component-wise:
(a,b)\cdot (a',b')=(a\cdot _A a',b\cdot _B b').

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

Subgroups

A subgroup B of A is a subset of A=(A,\cdot ) that is closed under the multiplication in A. (Notation: B\le A or B\leq A, the latter does not exlcude B=A.) This means that for any x,y\in B also x^{-1},xy\in B, and immediately implies 1\in B.

Examples of subgroups:

  1. Let G\leq S_n, and \Sigma\subset\Omega=\{1,\dots,n\}. Let H=\{h\in G\mid (\forall s\in\Sigma) h(s)\in\Sigma\}. Then H\leq G; it is called the stabiliser of \Sigma in G.
    Note that the stabiliser of \Sigma in S_n is isomorphic to S_m\times S_{n-m}, where m=|\Sigma|.
  2. Let G=GL_n(\mathbb{F}), and H=\{h\in G\mid \det(h)=1\}. Then H\leq G, is denoted by SL_n(\mathbb{F}), and is called the special linear group.
  3. Some subgroups of the direct product G=A\times B: let H\leq A, F\leq B. Then H\times F\leq G.

Generating sets, etc.
First, a bit of terminology: for a group A and any a\in A, n\in\mathbb{Z}, the power a^n is well-defined, as follows:

  1. if n=0 then a^n=1_A,
  2. if n<0 then a^n=(a^{-1})^{-n},
  3. if n>0 then a={a\cdot \dots\cdot a}, the n-fold product of a.

It can perfectly happen that a^n=1_A while neither n=0 nor a=1_A, e.g. a^{n!}=1_A for any a\in A=S_n.

Let S\subseteq A. Define \langle S\rangle, the subgroup generated by S, to be \langle S\rangle=\cap_{S\subseteq B\leq A} B, i.e. the intersection of all the subgroups of A containing S.
One can show that S consists of words in S\cup S^{-1}, i.e. finite products of the form
g_{i_1} g_{i_2}\dots g_{i_k}, where g_\ell\in S\cup S^{-1}, and we denoted S^{-1}:=\{s^{-1}\mid s\in S\}.

Cyclic subgroups. These are subgroups generated by just one element, s\in A, i.e. one should take S=\{s\} and denote \langle S\rangle=\langle s\rangle. We recall (just the notation changes from the additive to the multiplicative) that \langle s\rangle=\{s^n\mid n\in \mathbb{Z}\}.
More generally,

A group A is called finitely generated if there exists S\subset A such that \langle S\rangle=A and |S|<\infty.

E.g. any finite group, e.g. S_n, is finitely generated. E.g. GL_n(\mathbb{R}) is not finitely generated.

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