Endomorphism kernel is not transitive
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., endomorphism kernel) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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Contents
Statement
It is possible to have groups such that
is an endomorphism kernel in
,
is an endomorphism kernel in
, and
is not an endomorphism kernel in
.
Related facts
Similar facts about similar properties
- Normality is not transitive
- Complemented normal is not transitive
- Permutably complemented is not transitive
Opposite facts about similar properties
Combined facts with other properties
Related facts about endomorphism kernel
Proof
Further information: quaternion group, subgroup structure of quaternion group
Suppose is the quaternion group,
is a cyclic maximal subgroup, and
is the center of quaternion group. Explicitly:
Then:
-
is an endomorphism kernel in
: In fact,
is isomorphic to
.
-
is an endomorphism kernel in
: In fact,
is isomorphic to
.
-
is not an endomorphism kernel in
: In fact,
is isomorphic to the Klein four-group which does not occur as a subgroup of
.