Equivalence of definitions of fully invariant direct factor
From Groupprops
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 (?) of a group
:
-
is a Fully invariant subgroup (?) of
, i.e., every endomorphism of
sends
to itself. In other words,
is a fully invariant direct factor.
-
is a Homomorph-containing subgroup (?) of
, i.e., for any homomorphism of groups from
to
, the image of
is contained in
.
-
is an Isomorph-containing subgroup (?) of
, i.e.,
contains any subgroup of
isomorphic to
.