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ThomasBreuer merged 2 commits into from
Jan 29, 2026
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a little bit better documentation for nice monomorphisms #6211

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a little bit better documentation for nice monomorphisms
ThomasBreuer Jan 27, 2026
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ThomasBreuer Jan 27, 2026
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31 changes: 26 additions & 5 deletions doc/ref/grphomom.xml
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Expand Up @@ -344,11 +344,32 @@ nice monomorphism of a given group <C>G</C>,
for example if <C>G</C> is a matrix group and <C>map</C> describes the
faithful action of <C>G</C> on a small set of vectors.
In such a case, one can prescribe this monomorphism via
<C>SetNiceMonomorphism( G, map )</C>,
provided that <C>map</C> stores that it is injective.
The latter can be achieved by calling <C>IsInjective( map )</C> or,
if one is sure that <C>map</C> is injective and one wants to avoid the
overhead of the test, by calling <C>SetIsInjective( map, true )</C>.
<C>SetNiceMonomorphism( G, map )</C> if certain technical requirements
are satisfied.
<P/>
<List>
<Item>
<C>G</C> does not store a nice monomorphism yet.
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@james-d-mitchell james-d-mitchell Jan 27, 2026

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Suggested change
<C>G</C> does not store a nice monomorphism yet.
<C>G</C> does not store a nice monomorphism already (i.e. <C>HasNiceMonomorphism(G)</C> returns <K>false</K>).

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@ThomasBreuer ThomasBreuer Jan 27, 2026

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O.k.

</Item>
<Item>
<C>map</C> stores that it is injective.
This can be achieved by calling <C>IsInjective( map )</C> or,
if one is sure that <C>map</C> is injective and one wants to avoid the
overhead of the test, by calling <C>SetIsInjective( map, true )</C>.
Note that the <C>IsInjective</C> call for <C>map</C> may compute and store
some different nice monomorphism in the source of <C>map</C>,
which is probably <C>G</C>.
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@james-d-mitchell james-d-mitchell Jan 27, 2026

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So calling IsInjective may mean that you cannot call SetNiceMonomorphism after?

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@ThomasBreuer ThomasBreuer Jan 27, 2026

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You can call SetNiceMonomorphism for an object which has already a NiceMonomorphism stored, but this call has no effect, and you get no warning about this. (We do not check whether a second call of an attribute setter tries to set a different value, in the sense of =. First of all, this could be expensive. And there might also be logical problems; sometimes, equality would not be the right comparison because actually one needs object identity.)

An example where IsInjective actually sets a nice monomorphism is discussed in #6209.

</Item>
<Item>
<C>map</C> knows that it is a group homomorphism,
calling <Ref Oper="ImagesSet"/> with <C>map</C> and a subgroup of <C>G</C>
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@james-d-mitchell james-d-mitchell Jan 27, 2026

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Is there a way to check if this condition holds?

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@ThomasBreuer ThomasBreuer Jan 27, 2026

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I do not know an easy way, so I wanted to be careful with promises.

Initially I had thought that we can write that at least IsSPGeneralMapping is necessary for a nice monomorphism, but one might construct some map by hand, with its individual methods, where this does not hold but still it can be used as a nice monomorphism.

Usually something that makes sense as a nice monomorphism will be in IsGroupHomomorphism and IsInjective, but up to now we did not formally require this.
Technically, the point is that the images and preimages of group objects are again group objects, because this is assumed in the methods that transfer information along the mapping.

returns a group object, and analogously <Ref Oper="PreImagesSet"/> for an
image group returns a group object.
For example,
mappings constructed with <Ref Func="GeneralMappingByElements"/> will
<E>not</E> be <Q>nice enough</Q> to serve as nice monomorphisms.
</Item>
</List>

<#Include Label="IsHandledByNiceMonomorphism">
<#Include Label="NiceMonomorphism">
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4 changes: 3 additions & 1 deletion doc/tut/group.xml
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Expand Up @@ -1218,7 +1218,9 @@ composition of two maps (a nice monomorphism and an isomorphism)
which know that they are injective.
In general, the injectivity flag of a map <C>map</C> gets set if one tests
the property by calling <C>IsInjective( map )</C>;
if one is sure that <C>map</C> is injective and wants to avoid the
note that this call may compute and store some different nice monomorphism
in the source of <C>map</C>.
If one is sure that <C>map</C> is injective and wants to avoid the
potentially expensive test then one can set the flag by explicitly
calling <C>SetIsInjective( map, true )</C>.
<P/>
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