# Canonical examples of algebraic structures

Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.

"When I read about a [insert structure here], I immediately think of [example]."

Or maybe you think about a small number of examples. For example, when someone says "group", maybe you immediately think of one example of an abelian group and one example of a non-abelian group. Maybe you have a list of examples that you test new theorems on.

I am an analyst by training. When I read algebra, I can follow the logic line by line, but I don't have the repertoire of examples in my head that I have for analysis and so it's hard for me to picture anything.

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Neat question...

Abelian Group: Z or Z/nZ

Nonabelian Group: Dihedral groups or GL_n

Commutative Ring: Z or C[x]

Noncommutative Ring: Matrix rings

Division Ring: Hamilton's Quaternions

Field: R or C

Lie Algebra: sl_2

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I especially like sl_2 for lie algebra. I remember being amazed at how the representation theory of $sl_2$ gives you a good model to go about classifying all finite dimensional simple lie algebra over $\mathbb{C}$ –  Sean Tilson Nov 11 '10 at 1:53
@Sean: $sl_2$ is deceptively simple... I would say that $sl_3$ is a more representative example. –  Mariano Suárez-Alvarez Jun 20 '11 at 19:41
I can't believe somebody finds $\mathrm{GL}_n$ or dihedral groups more basic than $S_n$. –  darij grinberg Oct 27 '11 at 15:44
@darij: $S_n$ may be more basic, but $\operatorname{GL}_n$ is (to me) more transparent. I can see the structure: it contains $S_n$; it gives me examples of unipotent and solvable groups too; there's also a nice description (over $\mathbb{C}$) of the conjugacy classes. Also, of course, it reminds me of highest-weight theory for representations of reductive groups (even better than $\operatorname{SL}_n$, which is not fully reductive). I find the last one rather useful because of personal bias. Not as close to the axioms, maybe, but richer. –  Ryan Reich Oct 27 '11 at 16:49

Lots of good answers. I figured I'd throw in a list of non-examples, since these are pretty handy as well. (These are all standard non-examples, nothing fancy.)

A non-Noetherian ring with only one prime ideal: (k[x1, x2, x3, ...]/(xi xj : 1 <= i,j), (x1,x2,...)).

A non-Cohen-Macaulay ring: k[x, y]/(x2, xy).

A category that doesn't have products: the category of fields with field homomorphisms.

A ring which isn't flat over another ring: A = k[x2, x3] and B = k[x].

Two non-zero rings whose tensor product is zero: Z2 and Z3

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Vote up because one of my professors always says that following a definition, there should be an example and a non-example to reinforce understanding. –  B. Bischof Dec 1 '09 at 5:35

I often think of "universal examples". This is useful because then you can actually prove something in the general case - at least theoretically - just by looking at these examples.

Semigroup: $\mathbb{N}$ with $+$ or $*$

Group: Automorphism groups of sets ($Sym(n)$) or of polyhedra (e.g. $D(n)$).

Virtual cyclic group: Semidirect products $\mathbb{Z} \rtimes \mathbb{Z}/n$.

Abelian group: $\mathbb{Z}^n$

Non-finitely generated group: $\mathbb{Q}$

Divisible group: $\mathbb{Q}/\mathbb{Z}$

Ring: $\mathbb{Z}[x_1,...,x_n]$

Graded ring: Singular cohomology of a space.

Ring without unit: $2\mathbb{Z}$, $C_0(\mathbb{N})$

Non-commutative ring: Endomorphisms of abelian groups, such as $M_n(\mathbb{Z})$.

Non-noetherian ring: $\mathbb{Z}[x_1,x_2,...]$.

Ring with zero divisors: $\mathbb{Z}[x]/x^2$

Principal ideal domain which is not euclidean: $\mathbb{Z}[(1+\sqrt{-19})/2]$

Finite ring: $\mathbb{F}_2^n$.

Local ring: Fields, and the $p$-adics $\mathbb{Z}_p$

Non-smooth $k$-algebra: $k[x,y]/(x^2-y^3)$

Field: $\mathbb{Q}, \mathbb{F}_p$

Field extension: $\mathbb{Q}(i) / \mathbb{Q}, k(t)/k$

Module: sections of a vector bundle. Free <=> trivial. Point <=> vector space.

Flat / non-flat module: $\mathbb{Q}$ and $\mathbb{Z}/2$ over $\mathbb{Z}$

Locally free, but not free module: $(2,1+\sqrt{-5})$ over $\mathbb{Z}[\sqrt{-5}]$

... perhaps I should stop here, this is an infinite list.

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A more abstract version of "matrix ring" is the endomorphism ring of a module. If you take your module to be a free module over R, then you get matrix rings, but there are plenty of other examples of modules that are worth thinking about. This is my go-to example when I need a ring that isn't necessarily commutative.

There was another question asking for various examples of modules. Besides the free modules, the next-easiest R-modules for a ring R are ideals I and quotient rings R/I. In particular, remembering that abelian groups are Z-modules is useful.

The standard "geometric" principal ideal domain is k[X], for k a field. The standard "geometric" UFD is a polynomial ring over a field (or over a UFD). So if you want a UFD that isn't a PID, you have a bunch of choices, like k[X, Y] or Z[X]. If you want an integral domain that isn't a UFD, you can think of the coordinate ring of a generic affine variety.

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Projective module: given a vector bundle over a compact hausdorff space, the vector space of continuous sections of the bundle is a projective module over the algebra of continuous functions on the space.

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Not an algebraist myself, but this is interesting. Few things I do know:

algebraically closed field --------- $\mathbb C$

abelian group ---------------------- $(\mathbb R;+)$

non-abelian group ----------------- $(\operatorname{SO}(n,\mathbb R);\times)$ ; ($n>2$)

simple group ----------------------- $(A_n; \circ)$ ; ($n\ge5$)

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Wow, $\operatorname{SO}(n, \mathbb R)$ in preference to $S_3$? –  L Spice Mar 30 '10 at 15:30
$SO(2,\mathbb{R})$ is abelian. –  Pete L. Clark Mar 31 '10 at 4:59
What does it mean "(SO(n,R);x)"? The piece of notation that I don't understand is the "x". Same question for "(A_n;o)". –  Qfwfq May 11 '10 at 12:34
The "x" and "o" likely where meant to indicate the group "product", so "x"=$\times$ stands for "times" (product of matrices) and and "o"=$\circ$ for "composition" (of permutations), similarly to the "+" in (R;+) standing for regular addition. –  Max Horn Oct 14 '10 at 13:35
@Pete L. Clark and @unknowngoogle, I fixed the issues you mention. –  L Spice Jun 20 '11 at 15:56

More from the Lie-algebraic realm:

simple Lie algebra: of course, the ubiquitous sl(2)
solvable Lie algebra: < x,y | [x,y]=x >
nilpotent Lie algebra: < x,y,z | [x,y]=z, [x,z]=[y,z]=0 >
simple Lie algebra in characteristic p: < e_i, i\in Z/pZ | [e_i,e_j]=(j-i)e_{i+j} > (Witt algebra)
semisimple Lie algebra in characteristic p: sl(2)\otimes K[t]/(t^p) + Kd/dt
Kac-Moody algebra: sl(2)\otimes C[t,t^{-1}] + Ctd/dt + Cz

associative commutative algebra: K[t]/(t^n)

algebraic group: SL_2(K)

an algebraic object given by generators and relations: the corresponding 2-generated free object
(e.g., 2-generated free group, 2-generated free Lie algebra, etc.)

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I'll mention some more useful non-examples:

A non-monoid semigroup: $({\mathbb Z^+},+)$.

A non-group monoid: $({\mathbb N}, +)$, $({\mathbb Z},\cdot)$.

A non-integral domain: ${\mathbb Z}_6$.

A noetherian non-artinian ring: the integers ${\mathbb Z}$, the ring of Laurent polynomials over a field $K[x,x^{-1}]$.

A non-unital semisimple ring: the row-finite, column-finite, infinite matrices over a field ${\mathbb M}_{\infty}(K)$.

A simple non-semisimple algebra: The Weyl algebra $K[X,Y]/(XY-YX-1)$.

A non-simple indecomposable ring: $K[x,x^{-1}]$.

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Euclidean domain: Z[i] (Gaussian integers) Principal ideal domain: ring of integers in Q(\sqrt{-19}) Unique factorization domains: Z[x], C[x,y] Finite field: F_4=F_2[t]/(t^2+t+1)

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This is, at least, how it went for me just now:

Read:                Think:

field                Q(sqrt 2), or some other number field
finite group         symmetric group
Lie group            the torus R^2/Z^2
ring                 a matrix ring over... some other ring
commutative ring     a polynomial ring k[x_1,...,x_n]
Lie algebra          sl(2)

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This is a fun one! Here are some of my favorite examples (I'll try not to list ones that have already been listed).

Finite Group : SL_n(Z/pZ).

Nonabelian group (simple) : Free groups, Baumslag-Solitar groups

Nonabelian group (hard) : SL_n(Z), Aut(F_n), mapping class group

Lie algebra : Depending on context, either sl_n(C) or a free Lie algebra

Algebraic Variety : twisted cubic, a hyperelliptic curve

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The set of non-zero octonions is the canonical Moufang loop.

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$p$-group: the Sylow $p$-subgroups of $S_n$ or $GL_n(\mathbb{F}_p)$.

The former can be described as follows: split $n$ up into $\lfloor \frac{n}{p} \rfloor$ blocks of size $p$ and a remainder, and allow the permutations that only permute individual blocks. Then split the blocks themselves into "2-blocks" of size $p$ and allow permutations that permute the 2-blocks etc. The result is an iterated wreath product.

The latter can be described as the subgroup of upper-triangular matrices with all ones on the diagonal, e.g. a Heisenberg group.

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Is a $p$-Sylow subgroup of $\operatorname{GL}_n(\mathbb F_p)$ for $n \ne 2, 3$ still called a Heisenberg group? I tend to think of a Heisenberg group over $\mathbb F$ as being, in particular, in bijection with $V \times \mathbb F$ for some symplectic space $V$, so it has the wrong cardinality. –  L Spice Jun 20 '11 at 15:50
The other Sylow subgroups of $GL_n(\mathbb F_p)$ can be fun to describe too, at least in specific small examples. –  ACL Oct 27 '11 at 23:30

Pasha mentions presentations of groups; in this connection, I think there are few better examples than the Coxeter groups, with such beauties as $\langle a, b : a^2 = b^2 = 1, (ab)^3 = 1\rangle \cong S_3$.

In keeping with the theme of non-examples, I remember being puzzled by the notions of reductivity and quasi-split-ness until someone told me that a Borel subgroup of an algebraic group (like the group of invertible, upper-triangular $n \times n$ matrices) isn't the former, and the multiplicative group of a skew field (considered as an algebraic group over its centre) isn't the latter.

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I said this somewhere else before. But when people talk about projective limit, inverse limit, I think about $\mathbb{Z}_{p}$ as the projective limit of $\mathbb{Z}/p^{n}\mathbb{Z}$. When people talk about modular forms, which example do you usually think? Why?

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Modular forms are a little different. If you know what they are, you already know how to write a bunch of them down (at the very least, the Eisenstein series). My interpretation of the OP's question is that it applies to structures defined from a few simple axioms and properties where it's not clear how to find nontrivial examples of such things. –  Qiaochu Yuan May 11 '10 at 2:38
So you are probably thinking about Eisenstein series when someone mentions Modular forms but others may probably think about $\Theta$ function or $\Delta$ function. The definition of modular form sounds pretty artificial to me. No one ever told me when and why people start to study them. The question is irrelevant though. –  Haining Wang May 11 '10 at 4:25

The question title is about "Canonical examples of algebraic structures", but the text of the question (and most answers) make(s) me think I can probably answer in more generality to: "What do you do in front of a new (set of) definition(s)?", or "What do you take when a theorem asks to consider ...?".

I generally first check with too trivial examples, before I go to trivial but probably more interesting examples (as found in other answers) ; and in a third pass, I even consider objects which don't have one of the wanted properties (counter-examples, also discussed in other anwers). So let me tell more about that first step, since it was overlooked so far.

Of course, the defect is that for some situations you end up with an example so simple that it doesn't give anything. But those same correct but too primitive examples can later turn up as interesting and enlightening counter-examples in other definitions/results (I give two such situations below the list), and that gives them some merit.

Here is a short list of witless examples, to demonstrate how weak those first examples can/should be:

• A set: $\emptyset$ ;
• A monoid: {0} ;
• A group: {0} ;
• A ring: {0} ;
• A vector space: {0} ;
• An algebra: {0} ;
• An element in a structure : the zero or unit, if it fits the bill (if it doesn't, its time will come in the third step)
• A morphism: the zero map, if it fits the bill (if it doesn't... see above) or the identity ;
• A symmetry in an object: the identity (or -id) ;
• A projection in an object: the identity or the zero map ;
• A graded something: either grade with the zero ring or $\mathbb{Z}/2\mathbb{Z}$ ;
• A ring of integers in a number field: $\mathbb{Z}$ ;
• (not 100% algebraic) A distance on a space: the one which is zero on a diagonal pair and one outside ;
• (the same idea, but in a more algebraic setting) A valuation on a field: the trivial one ;

Here is a first example of the usefulness of being thickheaded: the definition of integral ring excludes the zero ring, and the definition of a prime element in a ring excludes the units. Both are linked to the question of the {0} ring, through the definition of a prime ideal by giving an integral quotient.

Now for a second such situation: finding an example of distance which doesn't come from a norm is trivial if your very first idea of a distance is the foolish one ; the same question is hard if you take honest examples from the second step, where you'll probably only have the usual suspects on $\mathbb{R}^2$.

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quote: "I generally first check with too trivial examples, before I go to trivial but probably more interesting examples (as found in other answers)" --- I found this a very helpful suggestion. –  Damodar Kulkarni Oct 29 '12 at 16:26

I think you'll find out very different my example of algebraic structures.

Monoid: the monoid of words over a finite alphabet.

Group: the group of words over a finite alphabet.

Ring: the polinomial ring.

Module: $\mathbb K^n$ for some $\mathbb K$ field.

Category: the path category over some graph $\mathbb G$.

$R$-Algebra (for some ring $R$): $M_n(R)$, i.e. the ring of matrix over the ring $R$.

These is exactly those algebraic structures that come in my mind when I think/try to prove fact about algebraic structures. Why? Because they're free object and are the terms models of their corresponding algebraic theories, so every other algebraic models derive from these by adding relations.

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