Twisted subgroup

This article defines a property of subsets of groups
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Definition

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.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subgroup				\|FULL LIST, MORE INFO
2-powered twisted subgroup	twisted subgroup within which every element has a unique square root.			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
1-closed subset	nonempty subset, contains the cyclic subgroup generated by any element in it			\|FULL LIST, MORE INFO
symmetric subset	nonempty subset, contains the identity and closed under taking inverses			\|FULL LIST, MORE INFO

Property theory

Associates

Further information: associate of twisted subgroup is twisted subgroup

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.