Base of a wreath product not implies elliptic

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., base of a wreath product) need not satisfy the second subgroup property (i.e., elliptic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about base of a wreath product|Get more facts about elliptic subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property base of a wreath product but not elliptic subgroup|View examples of subgroups satisfying property base of a wreath product and elliptic subgroup

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., 2-subnormal subgroup) need not satisfy the second subgroup property (i.e., elliptic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about 2-subnormal subgroup|Get more facts about elliptic subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property 2-subnormal subgroup but not elliptic subgroup|View examples of subgroups satisfying property 2-subnormal subgroup and elliptic subgroup

Statement

A base of a wreath product need not be an elliptic subgroup.

Proof

Further information: wreath product of Z and Z

In a wreath product of two infinite cyclic groups, the base cyclic group is a base of a wreath product but is not an elliptic subgroup.