Local finiteness is extension-closed
This article gives the statement, and possibly proof, of a group property (i.e., locally finite group) satisfying a group metaproperty (i.e., extension-closed group property)
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Statement with symbols
Suppose is a group with a normal subgroup such that both and are locally finite groups: any finitely generated subgroup of is finite, and any finitely generated subgroup of is finite. Then, is itself locally finite.
- Schreier's lemma: In its factual form, this states that any subgroup of finite index in a finitely generated group is again finitely generated.
- Finiteness is extension-closed
- First isomorphism theorem
Given: A group with a normal subgroup such that both and are locally finite. Let be the quotient map.A finite subset of with .
To prove: is finite.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||is a finite subgroup of||is locally finite||Since is finite, is finite. By local finiteness of , is a finite subgroup of . We have , so is finite.|
|2||is a normal subgroup of finite index in||Fact (3)||We have . Since the left side is finite, we conclude that has finite index in .|
|3||is finitely generated||Fact (1)||is finite||Step (2)||is finite, so is finitely generated, and by step (2), has finite index in . Thus, by fact (1), is also a finitely generated group.|
|4||is finite||is locally finite||Step (3)||is a finitely generated subgroup of . Local finiteness of forces to be finite.|
|5||is finite||Fact (2)||Steps (2), (4)||By step (4), is finite, and by step (2), is finite. Thus, by fact (2), is finite.|
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