Permuting upper join-closed subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
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VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

Definition

A subgroup property p is termed a permuting upper join-closed subgroup property if for any subgroup H of a group G and two intermediate subgroups K_1 and K_2 satisfying:

  1. H satisfies p in both K_1 and K_2, and
  2. K_1K_2 = K_2K_1, i.e., they are permuting subgroups,

we must have that H satisfies p in the join of subgroups \langle K_1 , K_2 \rangle which in this case is also the product of subgroups K_1K_2.

Relation with other metaproperties

Stronger metaproperties