RespectsMultiplication( mapp ) P
Let mapp be a general mapping with underlying relation
F Í S ×R,
where S and R are the source and the range of mapp, respectively.
Then RespectsMultiplication returns true if
S and R are magmas such that
(s1,r1), (s2,r2) Î F implies (s1 \* s2,r1 \* r2) Î F,
and false otherwise.
If mapp is single-valued then RespectsMultiplication returns true
if and only if the equation
s1^mapp * s2^mapp = (s1*s2)^mapp
holds for all s1, s2 in S.
RespectsOne( mapp ) P
Let mapp be a general mapping with underlying relation
F Í S ×R,
where S and R are the source and the range of mapp, respectively.
Then RespectsOne returns true if
S and R are magmas-with-one such that
( One(¢S¢), One(¢R¢) ) Î F,
and false otherwise.
If mapp is single-valued then RespectsOne returns true
if and only if the equation
One( S )^mapp = One( R )
holds.
RespectsInverses( mapp ) P
Let mapp be a general mapping with underlying relation
F Í S ×R,
where S and R are the source and the range of mapp, respectively.
Then RespectsInverses returns true if
S and R are magmas-with-inverses such that
(s,r) Î F implies (s-1,r-1) Î F,
and false otherwise.
If mapp is single-valued then RespectsInverses returns true
if and only if the equation
Inverse( s )^mapp = Inverse( s^mapp )
holds for all s in S.
Mappings that are defined on a group and respect multiplication and inverses are group homomorphisms. Chapter Group Homomorphisms explains them in more detail.
IsGroupGeneralMapping( mapp ) P
IsGroupHomomorphism( mapp ) P
A GroupGeneralMapping is a mapping which respects multiplication and
inverses. If it is total and single valued it is called a group
homomorphism.
KernelOfMultiplicativeGeneralMapping( mapp ) A
Let mapp be a general mapping.
Then KernelOfMultiplicativeGeneralMapping returns the set of all
elements in the source of mapp that have the identity of the range in
their set of images.
(This is a monoid if mapp respects multiplication and one, and if the source of mapp is associative.)
CoKernelOfMultiplicativeGeneralMapping( mapp ) A
Let mapp be a general mapping.
Then CoKernelOfMultiplicativeGeneralMapping returns the set of all
elements in the range of mapp that have the identity of the source in
their set of preimages.
(This is a monoid if mapp respects multiplication and one, and if the range of mapp is associative.)
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GAP 4 manual