Locally finite not implies embeddable in finitary symmetric group

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This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., locally finite group) need not satisfy the second group property (i.e., group embeddable in a finitary symmetric group)
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Statement

There can be a locally finite group that cannot be embedded as a subgroup of a finitary symmetric group.

Facts used

  1. Finitary symmetric group implies no non-identity element has arbitrarily large roots: see also group in which no non-identity element has arbitrarily large roots.

Proof

Example of the quasicyclic p-group

Consider the quasicyclic group for a prime p: the group of all (p^n)^{th} roots of unity in the complex numbers. This is clearly locally finite. On the other hand, it cannot be embedded in a finitary symmetric group, because every element can be expressed as a (p^n)^{th} power for arbitrarily large n, whereas no non-identity element of a finitary symmetric group can be expressed as a k^{th} power for arbitrarily large k.