Central factor is not quotient-transitive
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) not satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property).
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Contents
Statement
It is possible to have groups such that:
-
is a central factor of
. (In particular,
is normal in
).
-
is a central factor of
.
-
is not a central factor of
.
Definitions used
Central factor
Further information: Central factor
A subgroup of a group
is termed a central factor if
where
is the centralizer of
in
.
Note that any central subgroup, i.e., any subgroup contained in the center, is a central factor.
Related facts
- Centrality is not quotient-transitive
- Direct factor is quotient-transitive
- Central factor over direct factor implies central factor
Facts used
Proof
A generic example
Let be a finite non-Abelian group of nilpotence class two and
be maximal among Abelian normal subgroups of
. Consider
. Then:
-
is a central factor of
: In fact,
is a central subgroup of
-- it is contained in the center of
.
-
is a central factor of
: Since
is normal in
, we have
, and since
has class two, we have
. Thus,
. Thus,
is trivial, so
is central in
. Thus,
is a central factor of
.
-
is not a central factor of
: By fact (1),
is a self-centralizing subgroup of
. Since
is proper, this yields that
is not a central factor of
.
Example of the dihedral group
Further information: dihedral group:D8
Consider the dihedral group:
Define:
.
Then, we have:
-
is a central factor of
: In fact,
equals the center of
.
-
is a central factor of
: In fact,
is Abelian, so any subgroup of it is a central factor.
-
is not a central factor of
:
is a proper self-centralizing subgroup of
, so is not a central factor of
.
Example of the quaternion group
Further information: quaternion group
Consider the quaternion group, with identity element , and with the list of elements:
.
Define:
.
Then, we have:
-
is a central factor of
: In fact,
equals the center of
.
-
is a central factor of
: In fact,
is Abelian, so any subgroup of it is a central factor.
-
is not a central factor of
:
is a proper self-centralizing subgroup of
, so is not a central factor of
.