Twisted subgroup

Definition with symbols
A subset $$K$$ of a group $$G$$ is termed a twisted subgroup if it satisfies the following two conditions:


 * The identity element belongs to $$K$$
 * For every $$x \in K$$, $$x^{-1} \in K$$
 * Given $$x, y$$ in $$K$$, the element $$xyx$$ is in $$K$$

Note that the second condition is redundant when $$K$$ is a finite subset of $$G$$. Since twisted subgroups are usually studied in the context of finite groups, the condition is typically omitted from the definition. It is, however, necessary for the definition to behave nicely for infinite groups. The corresponding definition without this condition is better called twisted submonoid.

Associates
Let $$K$$ be a twisted subgroup of $$G$$. Then, for any $$a$$ in $$K$$, the sets $$Ka$$ and $$a^{-1}K$$ are equal and form another twisted subgroup. Such a twisted subgroup is termed an associate of $$K$$. The relation of being associate is an equivalence relation and we are interested in studying twisted subgroups upto the equivalence relation of being associates.

Intersection
See intersection of twisted subgroups is twisted subgroup.