Locally finite not implies embeddable in finitary symmetric group

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

Facts used

 * 1) uses::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.

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 $$k$$.