Help:Handling doubts and solving riders

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This is a first-person article written by: Vipul
First-person articles give opinions of their authors, as long as these opinions are broadly endorsed by the wiki

As a person new to group theory, or somebody doing a first course in the subject, one of the things you'll probably need is a resource that you can use to solve group theory problems that you are stuck with. There are many ways of doing this: talking to friends, posting problems on online fora such as Mathlinks, or going to teachers and asking them. Groupprops hopes to be an alternative solution for you. Right now, we are in a very preliminary stage of this effort.

A few things to bear in mind

Remember that Groupprops is not a guide or solution booklet or a kunji nor is it a discussion forum. Groupprops can offer to help because many of the things which you see as riders in textbooks are actually basic lemmas and cornerstones of the theory here. Further, many of the doubts that you have may be handled in the definition pages. But it is for you to formulate your doubt, to locate the relevant resources, and fit the information we have here into answers. It's an exercise in self-reliance.

If you get stuck and want some help, feel free to post your queries at the helpdesk for this wiki. If, however, you want to discuss homework problems that you are stuck with, the proper place to do this would be fora such as those on Mathlinks.

Handling a doubt in a definition

Reading and understanding the definition

First off, you can locate most of the basic definitions of group theory at:

Category:Basic definitions in group theory

Further, each page contains an explicit section with definition. Usually the definition is given in more than one way. For some terms, we have a symbol-free definition and a definition with symbols. For others, we have a textbook definition versus any other definition. Please try to match your definition against the correct one among these. Further, there are often pages describing why the multiple definitions are equivalent.

These pages are very useful in solving doubts, because many doubts are of the form this term was defined in this way here, and in that way there.

A problem identified with Groupprops definitions is that they are often too terse, and are not immediately accompanied by examples. To rectify and handle this problem, you can use the following tools:

  1. Read Groupprops:Definition to learn how to make best use of the definition section, including features like quick phrases.
  2. If there are multiple definitions, check out the definition equivalence page. This might give more insight into the definitions.
  3. To many of the basic definitions, there is a list of survey articles associated with that basic definitions. This list is linked to from the main page on the term. For instance Category:Survey articles related to normality lists the articles related to normality.
  4. For definitions of some popular terms, there are links to specific articles that go into the definition in more detail and explain it.

Finding other sources/references for the definition

  1. Many definition pages have a Textbook references subsection that lists various textbooks where the term is defined and explained, along with the page number and a brief description of the context in which the definition is introduced. Further information: Help:Using textbook references
  2. Many pages now have a "Searchbox" which gives a suggested search phrase and links to searches on that phrase in diverse resources like books. encyclopedias, discussion fora. Further information: Help:Using Searchbox
  3. There are also external links to definition pages on Wikipedia, Mathworld, Planetmath, and Springer Online References.

Handling a specific doubt

Doubts related to definitions could be of two types:

  • Not understanding the meaning of the definition
  • Not understanding the importance of the definition, in the sense of what an example could be, what a counter-example could be, and why have the definition at all

The first kind of doubt is a more serious one, and should hopefully be cleared up by reading and understanding the definition. If the definition you have seen does not match any of the definitions given on Groupprops, then one thing you could do is shelve your definition and accept the ones given on Groupprops, and let the matter clarify of its own accord.

Not understanding the importance or the motivation or not being able to construct examples or counterexamples is not such a serious handicap. Mathematicians have long played with structures they don't understand, with definitions for which they can't find examples, and so on. So if you have a doubt in these matters, a suggestion is -- shelve it, to be resolved later. Don't let it be a show-stopper to further reading. It's likely that the matter will be resolved in some time, particularly when you come across the right survey article or other informative piece or lecture notes.

Handling a rider

There are riders, homework assignments, and problem sets, which we often get stuck at. There are two approaches to these: try them all out on your own, or go post the problem on some discussion forum/ask somebody else for a solution.

Groupprops offers a middle solution: it's a kind of impersonal look-up point, which helps you around but not too much. Basically it's about relying on yourself to figure out where and how to locate the solution to your problem, by collecting relevant information on the terms related to the problem you are trying to solve, and about the important known facts on these terms.

Reading the definition pages of the terms

Homework problems have this habit of having lots of notation and so it's not easy to directly type the homework problem in the search bar (and we don't recommend you do that either). Instead, the first step you should do when you see a problem involving some term is to go back to the page for that term, and read the page thoroughly, scouring for information that may help you with that problem.

There are many advantages to this. Instructors often set homework problems and assignments to reinforce those concepts and ideas which they consider more important. Thus, it is likely that the number of problems on a particular term will be proportional to the extent to which that term is important. Further, it is also likely that each problem that refers to a concept and idea refers to it from a different angle.

Thus, if you refer to the Groupprops page each time the term is referenced, you will end up visiting and focussing on a different aspect. At the same time, you will also see the other aspects -- in particular, the definition, and thus all the ideas will sink in better.

In fact, it is recommended that you keep looking at the definition pages even if you already remember the definitions -- some new aspects may come up, or you may find some new and exciting links.

Figuring out whether the fact already exists

Again, the first place to get this idea is the main article for the term. For instance, consider the question:

Let H be a normal subgroup of G and K be any subgroup. Then prove that H \cap K is a normal subgroup of K.

Scouring the page on normal subgroup will lead to the subsection titled Transfer condition, which has exactly this as a statement. It also links to a proof of the statement, at Normality satisfies transfer condition.

Apart from using the definition page, one can also use listings of facts, such as Category:Basic facts in group theory and Category:Elementary non-basic facts in group theory.

Using textbooks

Further information: Help:Using textbook references

You can also try solving problems that occur in a standard textbook. Here are some ways:

  1. Type the name of the book, the problem number, and the page number in our internal search bar or when searching the site using Google or another search engine.
  2. Locate the page on the wiki about the book, in Category:Books.
  3. From this page, check out links to terminology defined, facts stated, facts proved, or all things referenced.

For instance, if you want to solve problem 9 on Page 70 of Herstein, you can type Herstein Page 70 Problem 9 in the searchbar. Of the five results you get, the fourth one, namely characteristic of normal implies normal, is exactly what you'd be looking for.

Note that this is not a primary, or recommended, use of the wiki. We aren't a solutions manual.

If it doesn't exist

Even if the fact and its proof don't exist, you can try to find related facts and proofs. Here are some tips on that:

  • Try changing some of the terms and search for related proofs. For instance suppose you want to see a proof that an intersection of characteristic subgroups is characteristic, but this proof isn't spelled out in the wiki. So you can instead look at the corresponding statement for normal subgroups, try to go over the proof, and see whether the proof can be adapted to characteristic subgroups.

Similarly, you could use ideas behind the proof that an intersection of normal subgroups is normal, to try proving that a join (subgroup generated) of normal subgroups is normal.