Periodic normal implies amalgam-characteristic
From Groupprops
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., periodic normal subgroup) must also satisfy the second subgroup property (i.e., amalgam-characteristic subgroup)
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Contents
Statement
Suppose is a group and
is a periodic normal subgroup of
: in other words,
is a normal subgroup of
as well as a periodic group (every element of
is of finite order). Then,
is an amalgam-characteristic subgroup of
:
is characteristic in
.
Related facts
Facts used
- Quotient of amalgamated free product by amalgamated normal subgroup equals free product of quotient groups
- Free product of nontrivial groups has no nontrivial periodic normal subgroup
- Normality is upper join-closed
- Normality satisfies image condition
Proof
Given: A group , a periodic normal subgroup
of
.
.
To prove: is a characteristic subgroup in
.
Proof:
- By fact (1),
.
-
has no nontrivial periodic normal subgroup: By fact (2), if
is a proper subgroup of
, then
has no nontrivial periodic normal subgroup. If
, then
is trivial and hence has no nontrivial periodic normal subgroup.
-
is a finite normal subgroup of
: Since
is normal in both copies of
, it is normal in
. Also,
is finite.
-
is the unique largest periodic normal subgroup of
: Suppose
is a finite normal subgroup of
. Then, by fact (4), the image of
in the quotient map
is a normal subgroup of
. Also, since
is periodic, its image is periodic. In step (2), we concluded that
has no nontrivial periodic normal subgroup. Thus, the image of
is trivial, so
.
-
is characteristic in
: This follows since it is the unique largest periodic normal subgroup of
.