# Groupprops:Guided tour for semi-beginners

*This article is a guided tour to Groupprops*

## Contents

## Whom this guided tour is for

This guided tour is for semi-beginners, viz people who:

- Have already done a first course in group theory (or equivalent)
- Are already familiar with the conventions, notations and ideas both within group theory and within closely related areas of mathematics (Such as ring theory, commutative algebra).
- Want to be able to get quick overviews and perform quick checks for group theory problems that may arise through different contexts.

## How to start

Since you already know a reasonable amount of group theory (at least the basic definitions part) you may choose to start in any of these ways:

- By performing a systematic exploration of the wiki
- By reading pages in and around the area where you are interested
- By starting off with the basic group theory pages (making sure that you remember the basics and that your language matches that in the wiki.

## Other places to check

You may also be interested in the following Resource pages:

- Resource:Online books in group theory
- Resource:Course materials in group theory
- Resource:Books in elementary group theory

## Exploring in and around your area of interest

### Getting started

Here's an example. Suppose you read the statement: *A subgroup is subnormal if and only if the iterated sequence of normal closures eventually terminates at the subgroup*. Now, you don't yet know what *subnormal* and *normal closure* mean (though you do have an idea what it means for a subgroup to be normal).

So the first thing to do is to check out the meanings of the words normal closure and subnormal subgroup. Both these meanings are stated clearly. For the term subnormal subgroup, multiple definitions are given, one of which is the above definition in terms of normal closures.

Now, since what we would like to understand is why the definition in terms of normal closures is equivalent to the other definition, the right place to look at is the equivalence of definitions of subnormal subgroup (incidentally, as of the time of writing the page, the equivalence page had not yet been created).

Here's another example. Suppose you read the statement *Every subgroup of a solvable group is solvable*. The right place to begin with the study of this statement is the article on solvable group. There, the relevant data, with reasons, is presented in the metaproperties section.

The idea typically is that even if it is difficult to locate the *fact* being referred to directly, one can refer to the *term*s involved in that fact, and hence try to locate the fact.

### Using categories to explore efficiently

For a complete listing of standard terms in group theory, refer:

Category: Standard terminology

For a narrower search, you can use the smaller categories:

- Category: Basic definitions in group theory should contain terms whose definitions you are already aware of. However, much of the rest of the information in the article may still be new to you.
- Category: Semi-basic definitions in group theory should contain terms whose definitions are fairly standard and are encountered at a level slightly beyond the rudimentary (they may be seen in a second group theory course or otherwise after completion of a first group theory course). This is the place where you will find the definitions of most of the terms you encounter.
- Category: Standard non-basic definitions in group theory should contain terms that are not commonly encountered in ordinary group theory texts, and although they may be standard, their use may not be common outside group theory. You may occasionally need to check out in this category.

Apart from exploring the terms based on their standardness and their level of difficulty, you could also try exploring them based on the qualitative kind of term it is. This is briefly described in the next section.

## Property-theoretic organization

This is just a preliminary discussion. A fuller and more comprehensive description of the way this wiki is organized is available at Groupprops:Organizational principles.

### The idea of property spaces

For ease of use and navigation, I have organized the wiki based on a property-theoretic paradigm. The idea is to club together all the group properties, all the subgroup properties, all the operators that act on groups etc. into relevant categories.

Even if you don't understand the global organization of the wiki, this should not be a deterrent to effectively exploring the wiki. Much of the structure is self-evident. Moreover, even if you do *not* follow the structure, tools like global search can always be used.

The typical things that you will need to be familiar with are:

- Group property: A group property is something that, given any group, is either true or false for that group. For instance, being Abelian is a group property because every group is either Abelian or not Abelian.

Typically, terms of the form *some adjective* followed by the word *group* describe group properties. However, this may not always be the case -- sometimes they may describe occurrence of groups or groups with additional structure. Still, a good first place to check when seeing a term of this kind is under the list of group properties.

A complete listing of the group properties on this wiki is found at:

Exploration and lookup pages for group properties will also be put up.

- Subgroup property: A subgroup property is something that, given any group-subgroup pair, is either true or is false for that group-subgroup pair. For instance, being normal is a subgroup property because every subgroup is either normal or is not normal.

Typically terms of the form *some adjective* followed by the word *subgroup* describe subgroup properties. There may also be other terms that describe subgroup properties, for instance, the member noun form (for instance central factor) or the property noun form.

A complete listing of subgroup properties on this wiki is found at:

Lookup and exploration guideline pages are also available.

- Automorphism property: An automorphism property is something that, given any group and an automorphism on that group, is either true or false for that automorphism on that group. For instance, the property of being inner is an automorphism property.

Typically terms of the form *some adjective* followed by the word *automorphism* denote automorphism properties.

### The idea of metaproperty

A metaproperty is a *property of a property*, viz it is something that can be evaluated over a property. Metaproperties are important because they can be used to give neat formulations of results involving properties. For instance, the fact that a characteristic subgroup of a characteristic subgroup is characteristic can be encoded by saying that the property of being characteristic satisfies the metaproperty of being transitive.

Similarly, the fact that an intersection of normal subgroups is normal can be encoded as saying that the subgroup property of being normal satisfies the metaproperty of being intersection-closed.

Thus, to be able to effectively understand properties and facts surrounding them, a mastery of the common metaproperties becomes helpful.

The following have been done to aid this mastery:

- The article on every group property and on every subgroup property contains a section titled
**Metaproperties**where the behaviour with respect to important metaproperties is discussed, along with links to the page for the metaproperty as well as a link to the proof. - The collection of metaproperties has a separate listing in a category.
- For each metaproperty, there is a category devoted to those properties that satisfy the given metaproperty.

### The idea of operator and property modifier

The normal core is an example of subgroup operator; so is the normal closure and normalizer. Thus, these are all listed under Category: Subgroup operators. We have similar notions of subgroup-to-group operators, group operators and so on.

In addition to operators that work directly on groups, there are also operators that work on subgroups. Some of these may be of interested to you as they may arise naturally in various situations. For instance, the subordination operator takes in a subgroup property and returns the property of being a subgroup for which a finite ascending chain can be fit from the subgroup to the group where each satisfies the property in its successor.

The listings of these operators are at Category: Subgroup property modifiers.

## Some nice sequences

If you really don't have a clue on how to begin, here are some nice suggested sequences:

### First round

- Check out the definition of subnormal subgroup, normal closure and related terms (figure out the related terms through links and categories)
- Check out the definition of characteristic subgroup, fully characteristic subgroup and strictly characteristic subgroup
- Look at Category: Subgroup-defining functions for the list of subgroup-defining functions.
- Check out the definition of simple group and then look at Category:Variations of simplicity for some interesting variations.
- Check out Category:Variations of normality and Category:Opposites of normality

### Second round

- Check out what is meant by a transitive subgroup property, and how it relates to the composition operator. Look at the proof of the fact that the subgroup property of being characteristic is transitive, and also that every characteristic subgroup of a normal subgroup is normal.
- Look at the proof that an arbitrary intersection of normal subgroups is normal. It is mentioned that this arises from the fact that normality is an invariance property. Check out the meaning of invariance property and the various other examples. Do something similar for the fact that an arbitrary join of normal subgroups is normal.\

(More rounds will be filled in later)

## Hazards

`Further information: Groupprops:Hazards`

### Temporary hazards

- Factual errors: Since all the articles put up have not been checked for errors, there may be some minor errors in the proofs and statements. Please do
*not*use this wiki as an authoritate knowledge source and use it only to point to further directions.

- Incomplete portions: Many of the definitions and proofs that are necessary to make the wiki reasonably complete, have either not been started, or have been left in-between. This problem is temporary and we finally hope to get all the portions at the semi-beginner level up to a reasonable degree of completeness.

### Long-term hazards

- There is a lot of terminology
*local to the wiki*that is mixed up with the standard terminology. As far as possible, I have stated clearly when a particular terminology is local to the wiki; however, it may still be confusing and overwhelming to see a lot of strange nad unfamiliar terminology thrown at one. Having the terminology mingled in, however, is necessary for other reasons.