Finite 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., finite normal subgroup) must also satisfy the second subgroup property (i.e., amalgam-characteristic subgroup)
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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finite group) must also satisfy the second group property (i.e., always amalgam-characteristic group)
View all group property implications | View all group property non-implications
Get more facts about finite group|Get more facts about always amalgam-characteristic group
Contents
Statement
Verbal statement
Any finite normal subgroup (i.e., a normal subgroup that is finite as a group) is an amalgam-characteristic subgroup.
Equivalently, a finite group is an always amalgam-characteristic group: it is an amalgam-characteristic subgroup in any group in which it is a normal subgroup.
Statement with symbols
Suppose is a finite normal subgroup of a group
. Then,
is a characteristic subgroup inside the amalgam
.
Related facts
Stronger facts
Similar facts
- Central implies amalgam-characteristic
- Periodic normal implies amalgam-characteristic
- Normal subgroup contained in hypercenter is amalgam-characteristic
Opposite facts
- Normal not implies amalgam-characteristic
- Characteristic not implies amalgam-characteristic
- Direct factor not implies amalgam-characteristic
- Cocentral not implies amalgam-characteristic
Applications
- Finite normal implies potentially characteristic
- Finite normal implies image-potentially characteristic
Facts used
- Normality is upper join-closed
- Quotient of amalgamated free product by amalgamated normal subgroup equals free product of quotient groups
- Free product of nontrivial groups has no nontrivial finite normal subgroup
- Normality satisfies image condition
Proof
Given: A group , a finite normal subgroup
of
.
.
To prove: is a characteristic subgroup in
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() ![]() |
Fact (1) | ![]() ![]() |
Since ![]() ![]() ![]() ![]() | |
2 | ![]() |
Fact (2) | ![]() ![]() |
Fact-direct | |
3 | ![]() |
Fact (3) | Step (2) | [SHOW MORE] | |
4 | ![]() ![]() |
Fact (4) | Steps (1), (3) | [SHOW MORE] | |
5 | ![]() ![]() |
Step (4) | [SHOW MORE] |
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