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advantage of the various permutations and permutation groups associated with
quasigroups. This chapter explains in detail how the permutations associated
with a quasigroup are calculated, and it also describes some of the core
methods of
in Chapter
in
the multiplication table of
it is a subquasigroup of
attribute
quasigroup
Let
of
understand whether the positions are meant among the elements of
the elements of the parent
obtain
quasigroup in
its Cayley table, are permanently tied to its parent.
must have entries covering the entire interval
same quasigroup), returns the corresponding list of positions of
elements of
=
generating sets of
loop)
containing
the positions of the respective elements in the parent quasigroup (resp.
loop) of
quasigroup, and it returns the one-element subloop of
(although not in
situations, for instance when the user calls
with empty center.
loop. Since there is no difference in the outcome while calling
call
(resp. loop)
and
a subset of
the same way as the elements of
consists of the list of first elements from the right cosets
obtained by
subset of
permutation of
their elements are permutations in the sense of
bijections of some interval
the numerical entries of the permutations point to the positions among
elements of the parent of
(resp. right translation) by
construction and the relationship between a quasigroup and its parent.
demand. Also note that changing the names of elements of a subquasigroup
(subloop) automatically changes the names of the elements of the parent
subquasigroup (subloop). This is because the elements are shared.
multiplication group of a quasigroup
multiplication group, resp. the relative multiplication group of a
quasigroup
multiplication group
multiplication group
analogously.
all
group is generated by all
mappings
group generated by all conjugations.
resp. the middle inner mapping
middle inner mapping group of a loop
quasigroup
subquasigroups (resp. subloops). When
fact groups, and they are returned as associative loops.
have therefore used
abbreviation is sometimes used in the literature, too.
associative loop.
containing all elements α α
a left inner mapping of
mappings of
normal subloop of
projection from
otherwise.
nilpotent.
else returns
else returns
strongly nilpotent loops.
quasigroups are isomorphic,
where
of
are not indexed by consecutive integers.
isomorphic,
of representatives of isomorphism classes of quasigroups from
representatives of isomorphism classes of loops from
convention on permutations as in
from
the elements of the underlying quasigroup without changing the isomorphism
type of the quasigroups.
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returns the quasigroup defined on the same set as
multiplication
returns the loop defined on the same set as
isomorphism
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of
cosets of
the top left corner of the Cayley table of the resulting loop.
calculate some loop invariants preserved under isomorphisms, and then we use
these invariants to partition the loop into blocks of elements preserved
under isomorphisms. The following two operations are used in the search.
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discriminator has been changed from version 3.2.0 up to accommodate
isomorphism searches for quasigroups.
have identical discriminators, they may or may not be isomorphic.
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isomorphism searches.
isotopic. The method works as follows: It is well known that if a loop
isotopic to a loop
that
of
isomorphic to
as a triple of bijections on
representatives of isotopism classes of loops from
