In the literature about
sporadic groups wordings like „ is involved in “[1] can be found with the apparent meaning of „ is a subquotient of “.
As in the context of subgroups, in the context of subquotients the term trivial may be used for the two subquotients and which are present in every group .[citation needed]
A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e. g.,
Harish-Chandra's subquotient theorem.[2]
Example
There are subquotients of groups which are neither subgroup nor quotient of it. E. g. according to article
Sporadic group, Fi22 has a double cover which is a subgroup of Fi23, so it is a subquotient of Fi23 without being a subgroup or quotient of it.
Order relation
The relation subquotient of is an
order relation – which shall be denoted by . It shall be proved for groups.
Reflexivity: , i. e. every element is related to itself. Indeed, is isomorphic to the subquotient of .
Antisymmetry: if and then , i. e. no two distinct elements precede each other. Indeed, a comparison of the group orders of and then yields from which .
The
preimages and are both subgroups of containing and it is and because every has a preimage with Moreover, the subgroup is normal in
As a consequence, the subquotient of is a subquotient of in the form
Relation to cardinal order
In
constructive set theory, where the
law of excluded middle does not necessarily hold, one can consider the relation subquotient of as replacing the usual
order relation(s) on
cardinals. When one has the law of the excluded middle, then a subquotient of is either the
empty set or there is an onto function . This order relation is traditionally denoted
If additionally the
axiom of choice holds, then has a one-to-one function to and this order relation is the usual on corresponding cardinals.