A-group

Definition

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 <math>G</math> if for any <math>p</math>-Sylow subgroup <math>P</math> of <math>G</math>, <math>P</math> is an Abelian group.

Metaproperties

Direct product

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.

Subgroups

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

Quotients

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.