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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
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Definition with symbols
Relation with other properties
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
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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.
This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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Any subgroup of an A-group is an A-group. This follows from the fact that a -Sylow subgroup of a subgroup is a -group in the whole group, and hence is contained in a -Sylow subgroup of the whole group, which is Abelian. Hence, the -Sylow subgroup of the subgroup is also Abelian.
This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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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.