Basic examples of groups

Contents

Cyclic groups
Dihedral groups
Number systems
The symmetric groups
The alternating groups
Free groups
Linear groups
The quaternion group
The Heisenberg group

This page contains descriptions of a number of groups that can be used as tests for the truth or falsity of general group-theoretic statements.

This article could use additional contributions. There are undoubtedly more examples that should be included. Also, the purpose of the explanations is to give the reader an idea of what kinds of assertions the groups can give examples of or counterexamples to: some of these explanations can definitely be improved.

Cyclic groups

The cyclic group of order $n$ , often denoted by $C_n$ , can be described in various ways. It is the group of all integers mod $n$ under addition, or the group of all rotational symmetries of a regular $n$ -gon, or the group of all $n$ th roots of unity under multiplication. An abstract definition of a cyclic group is that it is generated by one element: that is, a group $G$ is cyclic if there is an element $a\in G$ such that every element of $G$ is a power of $a.$

If $p$ is prime, then by Lagrange's theorem the cyclic group of order $p$ has no subgroups apart from two trivial ones: the identity and the whole group. It also follows from Lagrange's theorem that every group of order $p$ is cyclic (since every element $a$ generates a subgroup, so if $a$ is not the identity then that subgroup must be the whole group). Therefore, up to isomorphism there is exactly one group of order $p$ .

Since $C_p$ has no non-trivial subgroups, it is in particular a simple group: that is, it has no non-trivial normal subgroups. The groups $C_p$ are the only Abelian simple groups.

The group $\mathbb{Z}$ of all integers under addition is cyclic (according to the abstract definition, even if it is shaped more like a line than a circle). It is the only infinite cyclic group. Every subgroup of $\mathbb{Z}$ apart from $\{0\}$ is isomorphic to the whole group.

Dihedral groups

The dihedral group of order $2n$ , often denoted by $D_{2n}$ but also, slightly confusingly, often denoted by $D_n$ – we shall write $D_{2n}$ in this article – can be described concretely as the group of all symmetries of a regular $n$ -gon, or abstractly as the group generated by two elements $\rho$ and $\sigma$ that satisfy the relations $\rho^n=\sigma^2=1$ and $\sigma\rho\sigma=\rho^{-1}.$ (See presentations of groups if you do not understand the previous sentence.)

The dihedral group of order $6$ is isomorphic to $S_3,$ the group of permutations of a set of size $3.$ It is the only non-Abelian group of order $6,$ and the smallest non-Abelian group.

Dihedral groups are not simple, since the element $\rho$ generates a subgroup of index $2,$ and all subgroups of index $2$ are automatically normal. (It is also easy to see directly that this particular subgroup is normal.)

Number systems

If you want to know whether there is a group $G$ with a certain property, and $G$ is allowed to be infinite, then the most obvious examples to try are $\mathbb{Z}$ , $\mathbb{Q}, \mathbb{R}$ and $\mathbb{C}.$ You could also consider taking quotients of these; for example, $\Q / \Z$ is an infinite torsion group (i.e., every element has finite order).

The symmetric groups

A permutation of a set $X$ is a bijection from $X$ to $X.$ The symmetric group on a set of size $n$ , denoted $S_n$ is the group of all permutations of the set $\{1,2,\dots,n\}$ (or of any other set $X$ of size $n$ if that is more convenient).

Every group of order $n$ is isomorphic to a subgroup of $S_n.$ To see this, let $G$ be a group of order $n$ and associate with each element $g\in G$ the permutation $\pi_g$ from $G$ to $G$ that takes $h$ to $gh.$ The map $g\mapsto\pi_g$ is easily seen to be a homomorphism. Indeed, $\pi_{g_1}\pi_{g_2}(h)=\pi_{g_1}(g_2h)=g_1g_2h=\pi_{g_1g_2}(h).$

$S_4$ is isomorphic to the group of rotations of a cube (or regular octahedron). A good way to see this is to note that every rotation of the cube permutes the four diagonals of the cube. This gives us a homomorphism from the rotation group of the cube to $S_4.$ One can check that every rotation that sends each diagonal to itself is the identity, so the kernel of the homomorphism is trivial. Since there are $24$ rotations of the cube (each vertex has eight choices of where to go, and each neighbour of that vertex then has three choices of where to go, after which the rotation is determined), the homomorphism is an isomorphism.

The alternating groups

The alternating group on a set of size $n$ , denoted $A_n$ , is the group of all even permutations of the set $\{1,2,\dots,n\}$ (or, again, of any other set $X$ of size $n$ ).

$A_2$ is the trivial group and $A_3$ is the cyclic group of order $3$ , so the first interesting alternating group is $A_4,$ which has order $12.$ The subgroup of $A_4$ consisting of the identity and the three permutations $(12)(34), (13)(24)$ and $(14)(23)$ is normal. This follows because if $\rho$ and $\sigma$ are permutations, then $\rho\sigma\rho^{-1}$ has the same cycle type as $\sigma$ . Therefore, this subgroup is a union of conjugacy classes. However, for $n\geq 5$ it can be shown that $A_n$ is a simple group: the alternating groups form one of the infinite families of simple groups.

$A_5$ is isomorphic to the group of rotations of a regular dodecahedron. A nice way to see this is to start by observing that the vertices of the dodecahedron can be partitioned into (the vertices of) five regular tetrahedra. Therefore, $A_5$ acts on the set of these tetrahedra, which gives us a homomorphism from group of rotations of the dodecahedron to $S_5.$ One can show that any rotation that sends each of the five tetrahedra to itself must be the identity, so the kernel of this homomorphism is trivial. It is simple to prove that there are $60$ rotations of a dodecahedron (each face can to to one of twelve others, and can be rotated in one of five ways when it gets there), so the rotation group is isomorphic to a subgroup of $S_5$ of index $2.$ The only such subgroup is $A_5.$

Free groups

The free group on $k$ generators $x_1,\dots,x_k$ is the group whose elements are all strings $u_1u_2\dots u_n,$ where each $u_i$ is equal to $x_j$ or $x_j^{-1}$ for some $j.$ Two such strings are regarded as the same if you can get from one to the other by inserting or deleting pairs of the form $x_jx_j^{-1}$ or $x_j^{-1}x_j.$ For example, in the free group on the three generators $a, b$ and $c,$ the string $abb^{-1}a^{-1}cacc^{-1}$ represents the same element as the string $cab^{-1}b.$

If you are looking for a group generated by $k$ elements and you want it not to have any properties that are not forced on you by the group axioms, then the free group on $k$ generators is a good choice. Formally, this is expressed via a universal_property as follows. Let $F_k$ be the free group on the generators $x_1,\dots,x_k.$ Then, given any group $G$ and any $k$ elements $g_1,\dots,g_k$ of $G,$ there is a unique homomorphism $F_k\rightarrow G$ that sends $x_i$ to $g_i$ for each $i.$ It is not hard to see intuitively why this is true. Indeed, once one has mapped $x_i$ to $g_i,$ then the rest of $\phi$ is forced: for example, $x_1x_2^{-1}$ must map to $g_1g_2^{-1}.$ The only thing that is not quite trivial is that different strings that represent the same element of $F_k$ map to the same element of $G.$ But this follows from the fact that the only equations satisfied by $x_1,\dots,x_k$ are those guaranteed to hold by the group axioms, which means that they are also satisfied by $g_1,\dots,g_k.$

The ''free Abelian group on $k$ generators" is isomorphic to $\mathbb{Z}^k.$ The simplest set of generators of $\mathbb{Z}^k$ is the standard basis of $\mathbb{R}^k.$ The only equations satisfied by the generators are those guaranteed to hold by the group axioms and commutativity. This too can be expressed as a universal property – just replace "for any group" by "for any Abelian group" in the universal property for the free group $F_k.$ See also the Wikipedia article on free groups.

Linear groups

Many interesting groups arise as groups of invertible linear maps. An obvious example is the general linear group GL $(n,\mathbb{R})$ of all invertible linear maps from $\mathbb{R}^n$ to $\mathbb{R}^n,$ or equivalently of all invertible $n\times n$ real matrices. There are two ways of modifying this example: one can replace $\mathbb{R}$ by another field, and one can impose restrictions on the invertible maps. For instance, the special linear group SL $(n,\mathbb{R})$ is the group of all $n\times n$ real matrices with determinant $1,$ the orthogonal group O( $n$ ) is the group of all real orthogonal matrices, SL $(n,\mathbb{F}_p)$ is the group of all $n\times n$ matrices of determinant $1$ with elements in the field $\mathbb{F}_p$ with $p$ elements, and so on.

The group PSL $(n,\mathbb{F}_p)$ is the quotient of SL $(n,\mathbb{F}_p)$ by the subgroup of all scalar multiples of the identity. Except for a few very small values of $n$ and $p,$ PSL $(n,\mathbb{F}_p)$ is simple.

Linear groups over finite fields tend to have good expansion properties: if you choose a set $S$ of generators then the size of the set of elements you can form as a product of $k$ elements of $S$ grows rapidly as $k$ grows.

The quaternion group

The quaternion group is a group of order $8$ with elements $\pm 1, \pm i, \pm j, \pm k$ , which satisfy the relations $i^2=j^2=k^2=-1, ij=-ji=k, jk=-kj=i, ki=-ik=j,$ as well as the relations suggested by the notation, such as $(-1)^2=1,$ $(-1)i=-i,$ and so on.

The quaternion group is non-Abelian (since $ij\ne ji$ ). It is not isomorphic to $D_8,$ because it has only one element of order $2$ (namely $-1$ ), whereas $D_8$ has five (four reflections and a half turn).

The Heisenberg group

The Heisenberg group is the group of all $3\times 3$ matrices $\left(\begin{matrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{matrix}\right)$ with $a, b$ and $c$ belonging to $\mathbb{R}.$ The discrete Heisenberg group is the same but with $a, b$ and $c$ belonging to $\mathbb{Z}$ instead.

The Heisenberg group is non-Abelian, but it is close to Abelian, in the sense that it is $2$ -step nilpotent. The centre of the Heisenberg group is the set of all matrices for which $a=b=0,$ and the quotient by the centre is isomorphic to $\mathbb{R}^2$ (or $\mathbb{Z}^2$ in the discrete case). This makes the Heisenberg group a good group to begin with if one wishes to generalize a result that holds for Abelian groups.

Comments

Minimal examples

Wed, 27/05/2009 - 01:56 — Gabe Cunningham (not verified)

In the spirit of this page, would it be useful/desirable to have minimal examples of groups with certain properties (or failing to have certain properties?) Another similar idea would be to make a table using several basic group properties (e.g., abelian, simple, finite, torsion) and to give an example group or family of groups for each possible combination of properties. Perhaps such a table would be better served as part of a separate article detailing various nice properties of groups, with crosslinks to and from here. I'd be happy to start such a page if there is interest.

Articles along those kinds of

Sat, 30/05/2009 - 17:41 — gowers

Articles along those kinds of lines sound like a great idea to me.