Quotient-universal group property

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BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

Definition

A group property p is termed quotient-universal if given any group G, there exists a group K satisfying p and a normal subgroup N of K such that G is isomorphic to the quotient group K/N.

Note that the term quotient-universal is not standard, and some people use it in a narrower sense, for instance, only for finitely generated groups or only for finite groups.