The Group Properties Wiki (pre-alpha)

TIP: Learn how to better make use of textbook references

ABOUT US: We use a Creative Commons license. All our content is free to reuse, with attribution. Learn more

ALSO CHECK OUT: Commalg: The Commutative Algebra Wiki

From Groupprops

Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 2.3 Conjunction with other properties

BEWARE! This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this | Survey articles about this
Learn more about terminology local to the wiki | view a complete list of such terminology

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: | | | | |
RANDOM TIP:The metaproperties section lists important facts about the subgroup property, and addresses many of the natural questions that arise about it. It has links to proofs.

Definition

A subgroup H of a group G is termed homomorph-dominating in G if, for any homomorphism , there exists such that .

Relation with other properties

Stronger properties

• Order-dominating subgroup
• Homomorph-containing subgroup
• Sylow subgroup

Weaker properties

• Endomorph-dominating subgroup
• Isomorph-conjugate subgroup if the whole group is a co-Hopfian group -- it is not isomorphic to any proper subgroup of itself.

Conjunction with other properties

A homomorph-containing subgroup is precisely the same as a subgroup that is both normal and homomorph-dominating. For full proof, refer: Homomorph-dominating and normal equals homomorph-containing