A-group

Symbol-free definition
A finite group is termed an A-group if every Sylow subgroup of it is Abelian.

Definition with symbols
A finite group $$G$$ if for any prime $$p$$ dividing the order of $$G$$ and any $$p$$-Sylow subgroup $$P$$ of $$G$$, $$P$$ is Abelian.

Stronger properties

 * Abelian group
 * Z-group

Metaproperties
A direct product of A-groups is an A-group. This is because the Sylow subgroups of the direct product are the direct products of the individual Sylow subgroups.

Any subgroup of an A-group is an A-group. This follows from the fact that a $$p$$-Sylow subgroup of a subgroup is a $$p$$-group in the whole group, and hence is contained in a $$p$$-Sylow subgroup of the whole group, which is Abelian. Hence, the $$p$$-Sylow subgroup of the subgroup is also Abelian.

Any quotient of an A-group is an A-group. This follows from the fact that under a quotient mapping, the image of a Sylow subgroup remains a Sylow subgroup.