Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

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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Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Relation with other metaproperties 2.1 Weaker metaproperties

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This article is about a general term. A list of important particular cases (instances) is available at Category: Subgroup properties satisfying inverse image condition

Definition

Symbol-free definition

A subgroup property is said to satisfy the inverse image condition if, for any homomorphism of groups, the inverse image of a subgroup satisfying the property in the group on the right, satisfies the property in the group on the left.

Definition with symbols

Let p be a subgroup property. We say that p satisfies the inverse image condition if for any homomorphism , the following holds: whenever N is a subgroup satisfying p in H, φ ^{− 1}(N) satisfies p in G.

Relation with other metaproperties

Weaker metaproperties

• Intermediate subgroup condition
• Transfer condition
• Quotient-preserved subgroup property