# A-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

## 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 $G$ if for any prime $p$ dividing the order of $G$ and any $p$-Sylow subgroup $P$ of $G$, $P$ is Abelian.

## Metaproperties

### Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

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

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

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.

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

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.