Local finiteness is quotient-closed

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This article gives the statement, and possibly proof, of a group property (i.e., locally finite group) satisfying a group metaproperty (i.e., quotient-closed group property)
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Statement

Any quotient group of a locally finite group is also locally finite. In other words, if \varphi:G \to H is a surjective homomorphism, and G is locally finite, so is H.

Definitions used

Locally finite group

Further information: Locally finite group

A group is termed locally finite if every finitely generated subgroup of it is finite.

Related facts

Proof

Given: A locally finite group G, a surjective homomorphism \varphi:G \to H.

To prove: If A is a finite subset of H, \langle A \rangle is finite.

Proof:

  1. Construction of a finite set B \subseteq G such that \varphi(B) = A: Since \varphi is surjective, we can pick, for each a \in A, an element b \in G such that \varphi(b) = a. Making such a choice for each a \in A, we get a finite subset B of G such that \varphi(B) = A.
  2. \varphi(\langle B \rangle) = \langle \varphi(B) \rangle = \langle A \rangle: This follows from the fact that \varphi is a homomorphism.
  3. \langle B \rangle is a finite group (Given data used: G is locally finite): Since G is locally finite, and B is a finite subset, \langle B \rangle is a finite group.
  4. \langle A \rangle is finite: By steps (3) and (4), \langle A \rangle is the image of a finite group, and hence, is finite.