Solvable and generated by finitely many periodic elements not implies periodic

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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., solvable group generated by finitely many periodic elements) need not satisfy the second group property (i.e., periodic group)
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Statement

It is possible to have a group G that is a solvable group and is also a group generated by finitely many periodic elements, i.e., G has a generating set all of whose elements have finite order, but G is not a periodic group, and hence not a periodic solvable group.

This also shows that a solvable group generated by periodic elements need not be periodic.

Proof

Consider the infinite dihedral group:

\langle a,x \mid xax^{-1} = a^{-1}, x = x^{-1} \rangle

  • The group is solvable: It has an abelian normal subgroup \langle a \rangle with a quotient group that is abelian, isomorphic to cyclic group:Z2.
  • The group is generated by periodic elements: The elements x and ax form a generating set comprising elements of finite order (both elements have order two).
  • The group is not periodic: However, the element a in the group (which generates the cyclic maximal subgroup) has infinite order.