SQ-closed group property

Symbol-free definition
A group property is said to be SQ-closed if it satisfies the following equivalent conditions:


 * It is both subgroup-closed and quotient-closed.
 * Every subquotient of a group having the property also has the property.

Definition with symbols
A group property $$p$$ is said tobe SQ-closed if it satisfies the following equivalent conditions:


 * Whenever $$G$$ satisfies $$p$$, every subgroup $$H$$ of $$G$$ and every quotient $$G/N$$ of G</math< also satisfy $$p$$.
 * Whenever $$G$$ satisfies $$p$$, $$H$$ is a subgroup of $$G$$ and $$N$$ is a normal subgroup of $$H$$, then $$H/N$$ also satisfies $$p$$.

Weaker metaproperties

 * Subgroup-closed group property
 * Quotient-closed group property