Simple-complete subgroup property

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

Definition

Symbol-free definition

A subgroup property $p$ is said to be simple-complete if it satisfies the following:

• $p$ is trim, viz in every group, the trivial subgroup and the whole group satisfy property $p$
• Every group can be embedded in a group that is $p$-simple (that is, that satisfies the group property obtained by applying the simple group operator to $p$). In other words, every group can be embedded into a group that has no proper nontrivial subgroup satisfying $p$

Relation with other properties

Related properties

• Finite-simple-complete subgroup property: This is a trim subgroup property where every finite group can be embedded in a finite group which is simple with respect to that property