Groupprops:Definition

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

Formats of definition

Symbol-free versus with symbols

For most terms, the following two formats are used for specifying the definition:

• Symbol-free definition: This gives the definition without using any additional symbols. For instance, for permutable subgroup, the symbol-free definition states:

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 subgroup are not introduced.

The advantage of a symbol-free definition is that it sometimes brings out the essence without any symbol cluttering.

For a property operator, a symbol may be used for the property being operated on, to avoid overuse of pronous and to maintain some semblance of clarity.

• 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 $K$, $HK=KH$.

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:

• It is invariant under all inner automorphisms. Thus, normality is the invariance property with respect to the property of an automorphism being inner.
• It is the kernel of a homomorphism from the group.
• It equals each of its conjugates in the whole group.
• Its left cosets are the same as its right cosets.

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 $G$) if the following equivalent conditions hold:

• For all $g$ in $G$, $gNg^{-1}$ ⊆ $N$.
• There is a homomorphism $\phi$ from $G$ to another group such that the kernel of $\phi$ is precisely $N$.
• For all $g$ in $G$, $gNg^{-1}=N$.
• For all $g$ in $G$, $gN=Ng$.

Separate definitions as separate subsections

Sometimes, if the equivalent definitions are long and intricate or both require 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 labelled as the textbook definition and the universal algebraic definition.

Equivalence of mutiple 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. This can be seen, for instance,

Links from within the definition

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 first 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.