A-group: Difference between revisions

Latest revision as of 16:53, 14 December 2023

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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

Weaker properties

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.

@@ Line 1: / Line 1: @@
+{{group property}}
+{{semistddef}}
 ==Definition==
 ===Symbol-free definition===
-A [[finite group]] is termed an '''A-group''' if every [[Sylow subgroup]] of it is Abelian.
+A [[finite group]] is termed an '''A-group''' if every [[Sylow subgroup]] of it is abelian.
 ===Definition with symbols===
-A [[finite group]] &lt;math&gt;G&lt;/math&gt; if for any &lt;math&gt;p&lt;/math&gt;-[[Sylow subgroup]] &lt;math&gt;P&lt;/math&gt; of &lt;math&gt;G&lt;/math&gt;, &lt;math&gt;P&lt;/math&gt; is an [[Abelian group]].
+A [[finite group]] <math>G</math> if for any prime <math>p</math> dividing the order of <math>G</math> and any <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math>, <math>P</math> is [[abelian group|abelian]].
+==Relation with other properties==
+===Stronger properties===
+* [[Abelian group]]
+* [[Z-group]]
+===Weaker properties===
 ==Metaproperties==
-===Direct product===
+{{DP-closed}}
 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===
+{{S-closed}}
-Any [[subgroup]] of an A-group is an A-group. This follows from the fact that a &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of a subgroup is a &lt;math&gt;p&lt;/math&gt;-group in the whole group, and hence is contained in a &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the whole group, which is Abelian. Hence, the &lt;math&gt;p&lt;/math&gt;-Sylow subgroup of the subgroup is also Abelian.
+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===
+{{Q-closed}}
 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.
-[[Category: Finite group properties]]
+==Examples and counterexamples==
-[[Category: Group properties]]
+===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.