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]

VIEW: Definitions built on this | Facts about this: (facts closely related to A-group, all facts related to A-group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View other semistandard definitions in group theory

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.

Relation with other properties

Stronger properties

• Abelian group
• Z-group

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.

Examples and counterexamples

Examples

• Every abelian group is an A-group.
• symmetric group:S3 is the smallest non-abelian A-group. (Indeed, it is the smallest non-abelian group.)

Counterexamples

• The smallest groups that are not A-groups are the two non-abelian groups of order 8 (dihedral group:D8, quaternion group). Since 8 is a prime power, their Sylow subgroups are themselves, and hence are not abelian.
• More generally, a non-abelian group of order a prime power is not an A-group.