# Groupprops:Definition

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The Definition section in Groupprops should serve as a complete introduction to what the term means.

## Goals of the definition section

### Potential audience for a definition section

Some people know the term and are looking for its definition. This could include:

• People encountering the term for the first time
• People who've seen the term before, and know what it means, but want a precise definition
• People who already know one definition, but want to know/read multiple definitions

On the other hand, there could also be people who know the definition already, and are looking for the precise term having that definition.

### What the definition should provide

A definition should be clear, succinct, actionable, and interesting.

• It must clearly answer the question: what is this
• If the definition provides a differentiation or augmentation of an existing concept (for instance, a particular breed of dog, or a particular property of groups, or a particular type of number) then it should give a clear differentiating criterion.
• It should give an idea of what related terms and facts are and how one can explore the notion better
• It should be amenable to reverse search

## Formats of definition

### Quick phrases

Quick phrases are to be seen at the top of definition sections in some articles. These are phrases which one can use to remember the term being defined in a very easy way. For instance, a quick phrase for a prime number is number with no nontrivial factorization.

For quick phrases, conceptual ease matters more than precise meaning, so some of the quick phrases may not make precise mathematical sense.

### Symbol-free definition

A symbol-free definition is a definition that avoids that use of algebraic symbols and notation, relying instead on constructs of natural language such as pronouns and prepositions. For instance, here's a symbol-free definition of permutable subgroup:

A subgroup of a group is termed permutable if its product with every subgroup of the group is a subgroup, or equivalently, if it commutes with every subgroup.

Here, symbols for the group and the two subgroups ar enot introduced. Rather, constructs of natural language, in particular, the pronoun it, are used.

Sometimes 1-2 symbols may be introduced in a symbol-free definition to avoid ambiguous pronouns.

1. More effective for reverse search i.e. locating the term from the definition.
2. Might be easier to read if you already know the definition and are trying to recall it (because of less demand on working memory)
3. Easier to keep in mind, memorize, and act upon in more diverse situations where the notation and symbols differ
4. More effective linking with other concepts and ideas

### Definition with symbols

This gives the definition with explicit introduction of symbols for every quantified object. For instance the definition with symbols of the property of being a permutable subgroup is:

A subgroup $H$ of a group $G$ is termed permutable if for every subgroup $K$, $HK=KH$.

The definition with symbols has a number of advantages, for instance:

• It may be better for first-time reading
• It may be more actionable
• It may require less knowledge of the subject or of terminology in the subject

## Multiple definitions

### Equivalent definitions as a list

Often, the same term may have multiple equivalent definitions. In this case, all the equivalent forms are stated as bullet points. The same format is repeated in the symbol-free definition and in the definition with symbols.

For instance, the symbol-free definition of normal subgroup: A subgroup of a group is said to be normal if it satisfies the following equivalent conditions:

1. It is the kernel of a homomorphism from the group.
2. It is invariant under all inner automorphisms. Thus, normality is the invariance property with respect to the property of an automorphism being inner. This definition also motivates the term invariant subgroup for normal subgroup (which was used earlier).
3. It equals each of its conjugates in the whole group. This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier).
4. Its left cosets are the same as its right cosets (that is, it commutes with every element of the group)
5. It is a union of conjugacy classes.

The definition with symbols of normal subgroup:

A subgroup $N$ of a group $G$ is said to be normal in $G$ (in symbols, $N \triangleleft G$ or $G \triangleright N$Notations) if the following equivalent conditions hold:

1. There is a homomorphism $\phi$ from $G$ to another group such that the kernel of $\phi$ is precisely $N$.
2. For all $g$ in $G$, $gNg^{-1} \subseteq N$. More explicitly, for all $g \in G, h \in N$, we have $ghg^{-1} \in N$.
3. For all $g$ in $G$, $gNg^{-1} = N$.
4. For all $g$ in $G$, $gN = Ng$.
5. $N$ is a union of conjugacy classes

We try to follow these conventions:

• When multiple definitions are given, the same order is maintained for the definitions in the symbol-free format and the definition with symbols format.
• The multiple definitions should preferably be numbered, making it easier to refer to the numbering
• The definitions are ordered in a way that makes the equivalence between them as self-evident as possible.

### Separate definitions as separate subsections

Sometimes, if the equivalent definitions are long and intricate or each requires their own machinery, they may be developed in separate subsections. The title of each subsection gives some title to the definition.

For instance, the two definitions of group are labeled as the textbook definition and the universal algebraic definition.

### Equivalence of multiple definitions

Usually, there is a separate section within the definition part explaining the equivalence of definitions, or giving a link to a page that gives a full proof of this equivalence.

## Defining ingredients

Whenever a term is being defined using other terms, put links to those terms. For instance, when defining a group property, begin with: A group is said to be ... if ...

Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the second point states that a subgroup is normal if it is invariant under all inner automorphisms. In the definition with symbols, the explicit form of inner automorphisms is given.

### Locating defining ingredients

The wiki markup contains information about what terms are used as ingredients in defining a particular term (using semantic MediaWiki). At the bottom of the article, a box called Facts gives a list of defining ingredients. Clicking on the magnifying glass icon next to any of these ingredients gives a list of all the other terms that use the same defining ingredient.

### Links to survey articles clarifying the definition

For definitions of important terms, the definition section may begin with a box containing links to survey articles clarifying the definition.