Equivalence of definitions of fully invariant direct factor

From Groupprops
Revision as of 20:12, 11 August 2009 by Vipul (talk | contribs) (Created page with '{{definition equivalence|fully invariant direct factor}} ==Statement== The following are equivalent for a fact about::direct factor <math>H</math> of a group <math>G</m…')
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article gives a proof/explanation of the equivalence of multiple definitions for the term fully invariant direct factor
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a Direct factor (?) H of a group G:

  1. H is a Fully invariant subgroup (?) of G, i.e., every endomorphism of G sends H to itself. In other words, H is a fully invariant direct factor.
  2. H is a Homomorph-containing subgroup (?) of G, i.e., for any homomorphism of groups from H to G, the image of H is contained in H.
  3. H is an Isomorph-containing subgroup (?) of G, i.e., H contains any subgroup of G isomorphic to H.

Proof

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]