1-closed subset

Symbol-free definition
A nonempty subset of a group is termed a 1-closed subset if it satisfies the following equivalent conditions:


 * 1) It is a union of subgroups
 * 2) It is a power-closed symmetric subset
 * 3) For any element in it, the cyclic subgroup generated by that element is also in it. In particular, it is a union of cyclic subgroups.

Stronger properties

 * Subgroup
 * Twisted subgroup

Weaker properties

 * Symmetric subset
 * Power-closed subset

Related facts

 * Every group is a union of cyclic subgroups
 * Cyclic iff not a union of proper subgroups