Finite normal implies amalgam-characteristic

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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

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 H is a finite normal subgroup of a group G. Then, H is a characteristic subgroup inside the amalgam L := G *_H G.

Related facts

Stronger facts

Similar facts

Opposite facts

Applications

Facts used

  1. Normality is upper join-closed
  2. Quotient of amalgamated free product by amalgamated normal subgroup equals free product of quotient groups
  3. Free product of nontrivial groups has no nontrivial finite normal subgroup
  4. Normality satisfies image condition

Proof

Given: A group G, a finite normal subgroup H of G. L := G *_H G.

To prove: H is a characteristic subgroup in L.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 H is a finite normal subgroup of L. Fact (1) H is finite and normal in G Since H is normal in both copies of G, it is normal in L. Also, H is finite.
2 L/H \cong G/H * G/H. Fact (2) H is normal in G Fact-direct
3 L/H has no nontrivial finite normal subgroup. Fact (3) Step (2) [SHOW MORE]
4 H is the unique largest finite normal subgroup of L. Fact (4) Steps (1), (3) [SHOW MORE]
5 H is characteristic in L. Step (4) [SHOW MORE]
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format