# Twisted subgroup

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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
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.