# What is the importance of examples in the study of group theory?

When I study topics in group theory (I am currently following Dummit and Foote) I don't care about examples so much. I read them, try to understand the applications of the theorems and corollaries on the examples. Most of the examples are about $D_{2n}, S_n$ and $\mathbb{Z}_n$, etc.

However, if I have difficulty with them, sometimes I skip them (although I mostly try hard to understand them first). The same happens with exercises, I care more about questions which ask me to prove general statements - in the form of corollaries or theorems. For example: "prove that $G$ is abelian if $G/Z(G)$ is cyclic"

So, my question is, what is the importance of examples of group theory? Should I care more about examples? Or will I be okay if I understand the definitions, theorems, corollaries and solve the proof-based exercises well?

Also, is there a reference - a book or a collection of sites - which talks about the groups only? I mean it talks about, for example, $D_{2n}$, Matrix groups , $S_n$ , Klein four-group etc and the relations between those groups, their properties, how they act on quotients, the relation between the center and factor of the group and their subgroups, their normal subgroups, their sylow p-subgroups and so on.

If no such reference exists, then I have to collect the information of each group from the sections and exercises and try to summarize these information of each group which is mentioned in the text.

• Learning lots of examples of groups which are "weird" in different ways helps to understand the limitations of how groups can behave. It develops your intuition about what can and can't happen in the world of groups. Additionally, if you have a bunch of counterexamples on hand while investigating a question, you can perform sanity checks to prevent spending too much time going down the wrong path. If you'll excuse the self-plug, it may help to read this post, in which I explain the importance of some specific families of (finite) groups . – Alexander Gruber Jun 6 '13 at 4:50
• @Maths Lover, I have made an edit to your post in an attempt to make it easier to read. Please look over it and make sure I haven't accidently changed the meaning of your question. – Tom Oldfield Jun 6 '13 at 22:20
• @TomOldfield , great work :) my english is not perfect so errors happen always ! thanx for your corrections :) – Fawzy Hegab Jun 6 '13 at 23:28
• @MathsLover No problem, happy to help! – Tom Oldfield Jun 7 '13 at 0:20
• There is an old tale about a group theory PhD student who managed to prove many wild and wonderful properties about a class of groups. When he came to defend his thesis the examiner asked "okay, can you give me an example of a group from this class?" It soon transpired that the only possible example was the trivial group, and the thesis imploded under the pressure of tears, toil and broken dreams... Moral: Examples are important. – user1729 Jun 7 '13 at 8:27

There is some pedagogical value to understanding how theorems/definitions apply to and playout with particular groups (i.e., through examples). While it is admirable to yearn for a thorough theoretical understanding of group theory, try to think of studying and probing the examples as a means of testing your understanding of the theories and definitions, as they relate to specific groups. Being able to construct a proof is wonderful, but being able to use and apply your learning is crucial to obtaining a fluency with what you're learning. Being able to understand and apply the theorems and definitions is also a test of your comprehension, and will facilitate the proof-writing process as well. For example, being able to construct and anticipate counterexamples is crucial to proof-writing, and counterexamples will be hard to come by if you don't understand how particular groups exemplify or fail to exemplify group properties.

A great website is the website Groupprops - "A Group Properties Wiki". You'll find a menu to the left:

Terms/definitions
Facts/theorems
Survey articles
Specific information


and can quickly look-up (almost) anything/everything you'd want to know about (almost) any group.

• your answers always provide the material and things which i precisely look for ! that's why you are one of my heroes on maths.stackexchange . i explored the site a little , it seems to include huge amount of information! this will help me so much to learn more about new theorems , definitions and examples " one new definition which i found was the definition of order-statics - . are there particular groups which the student of group theory should feel comfortable with dealing with them ? – Fawzy Hegab Jun 6 '13 at 3:06
• @Maths Lover: Thank you very much! Let me think about your question for a little while. There are far too many groups to know them all well, but knowing families of groups is helpful: and knowing that groups of various sorts exist is half the battle (one can always turn to Groupprops, and the like, to refresh one's memory, and/or as the need arises). Dummit & Foote have chosen examples carefully, and so the examples presented are presented for a reason. Always ask yourself: "What is key/special about this group $G$ being presented now, in this context?" – Namaste Jun 6 '13 at 3:22
• Note what the group is being used to exemplify...that will help you understand both the concepts and the particulars of various groups, and will serve as a "mental bookmark" to attach to the exemplifier groups. Again, often knowing that such groups exist is half the battle... – Namaste Jun 6 '13 at 3:24

If you want to use group theory for anything, then I would at least recommend making the permutation and linear groups your friend, since they have a much broader place in mathematics. But in general you should probably learn to work with and create examples: the examples are the things that let you recognize where group theory will help you when you're off doing something else! Not to mention that proving that something isn't true is very difficult to do without an understanding of how to actually construct groups with specific properties (i.e. counterexamples).

could you, say:

-find a non-cyclic group whose subgroups are all cyclic and nested one inside the other?

-find a nonabelian group whose subgroups are all cyclic?

-show that $SL(n,\mathbb{R})$ is connected?

-find the symmetry groups of the platonic solids?

Questions like these are important and tell you a lot about the gifts group theory has brought to the rest of mathematics. The groups that answer these questions (and others like them) aren't just random examples, they pop up fairly often in other problems and can make excellent test cases for new ideas. While you could probably learn a lot of the theorems by themselves, I'd say you're missing out on a vast amount of the subject if you ignore the actual groups themselves.

Some very good mathematicians use geometric interpretations to help guide themselves through complicated ideas and proofs in algebra and other subjects. Just last night Erik Demaine gave a public lecture here in Halifax where he showed pictures, I tell you, of physical models that he built, out of paper, blown glass, and every other interesting thing he could find, apparently motivating and motivated by some mathematical problem he was thinking about.

Some students are helped a lot by geometric interpretations; others just want the rules of the game. However, if Dummit and Foote are trying to tell you a geometric interpretation of a group, then consider whether or not there might be a good reason for that, for example from a psychology and neuroscience perspective.

• I suppose the question was more about examples than geometric interpretations, however, the two points are related... geometric interpretations typically come in the form of concrete examples (especially when they are literally examples made out of concrete). – Douglas B. Staple Jun 6 '13 at 2:49
• If you are interested in physical models, you might be interested in this cool paper, which deals with geometry blah blah blah 3D printers. I have seen Saul Schleimer try (successfully!) to sell his models, although he billed them as "a kind of jigsaw puzzle" as they leave the printer in very small component parts... – user1729 Jun 7 '13 at 8:33

This post it about infinite groups because that is what I know about, but the motivations I talk about can still be applied to finite groups (such as the sporadic simple groups).

Examples help us to understand theoretical bounds on the properties of groups. For example, not all "recursively presented" groups have "soluble word problem". Basically, given a group this means that it is not necessarily possible to find out if two elements are actually the same. This is a fundamental result which was proven in the 50s, and shows that groups can be very nasty indeed. The proof was a direct construction of a recursively presented group with insoluble word problem.

Using this result, examples have been constructed which prove that, in general, it is not possible to determine if two groups are isomorphic!

The group $G=\langle a, b; b^{-1}a^2b=a^3\rangle$ possesses a surjective map $\theta: G\twoheadrightarrow G$ which is not injective. Which is crazy! (Such a group is called non-Hopfian.)

So, examples are important because they can tell that groups can have nasty properties. However, people often study a single example of a group, and sometimes this is because the group might be a counter-examples to lots of questions! For example, Thompson's group F is already a counter-example to many questions, and it has become an intriguing open problem as to whether this group is amenable or not. Either answer has repercussions.

Finally, examples can often serve as test cases for a larger class of groups. For example, in the 80s Gromov introduced a class of groups called "hyperbolic groups" and he asked if they were "residually finite". A specific sub-class of these, called "one-relator groups with torsion" (groups with presentation $\langle X; R^n\rangle$, $n>1$), have been studied since the 30s, and it was an open problem of G. Baumslag in the 50s as to whether these groups were residually finite. So these one-relator groups became a test-case, but really noone knew how to prove either result. Until Dani Wise came along last year and blew our minds. We now know that many, many hyperbolic groups are residually finite, and principle among these are one-relator groups with torsion. So the test case works. Yay!

You wrote:

"The same happens with exercises, I care more about questions which ask me to prove general statements - in the form of corollaries or theorems. For example: "prove that G is abelian if G/Z(G) is cyclic"

In any field of research mathematics, it is very rare that what you do is to follow someone elses instructions "prove this". Instead, what you do is to explore the field to try to discover true facts, and once you have convinced yourself of the truth of those facts you then try to prove them. The discovery process is, I think, literally impossible without knowing lots of examples, looking amongst them for patterns, and learning how to construct new examples and explore them too.

Also, from what you write it sounds like you prefer statements that have the logical structure "Prove that for all groups that satisfy such-and-such, they also satisfy so-and-so". Trouble could occur if you were to run into such a statement which was false. You would then be faced with proving a statement having a different logical structure: "Construct a group that satisfies such-and-such and does not satisfy so-and-so".

It would be more instructive to seek out exercises which are structured somewhat differently: "Is it true that all groups satisfying such-and-such also satisfy so-and-so? If so, give a proof. If not, give a counterexample." Questions with this structure portray research mathematics more realistically. But without having a hearty collection of examples in your pocket ready to use, you might have a hard time handling the "If not..." part of such questions.

• Great advices , i really try to develop this sense with exapmles , this surely will help me – Fawzy Hegab Jun 19 '13 at 9:25

From my experience as a student, I find that concrete examples (well, if anything in mathematics can be called "concrete") help me to understand the more abstract ideas from definitions and theorems. In my opinion, having examples at one's disposal deepens one's understanding of a given topic. It also provides a way to communicate your understanding, especially in a classroom environment.