text: # Finite implies powering-invariant
  
  This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., finite subgroup) must also satisfy the second subgroup property (i.e., powering-invariant subgroup)
  View all subgroup property implications | View all subgroup property non-implications
  
  ## Statement
  
  Suppose $G$ is a group and $H$ is a finite subgroup of $G$. Then, $H$ is a powering-invariant subgroup of $G$. Explicitly, if $p$ is a prime number such that $G$ is powered over $p$, then $H$ is also powered over $p$.
  
  ## Proof
  
  Given: A group $G$, a finite subgroup $H$, a prime number $p$ such that the map $x \mapsto x^p$ is bijective from $G$ to itself.
  
  To prove: The map $x \mapsto x^p$, restricted to $H$, is bijective from $H$ to itself.
  
  Proof:
  
  Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
  1 The map $x \mapsto x^p$ sends $H$ to within itself. $H$ is a subgroup.
  2 The map $x \mapsto x^p$ (restricted to $H$) is injective from $H$ to itself. The map $x \mapsto x^p$ is bijective from $G$ to itself. Step (1) The map is bijective on $G$, hence injective on $G$, hence its restriction to $H$ must also be injective.
  3 the map $x \mapsto x^p$ (restricted to $H$) is injective from $H$ to itself. $H$ is finite. Step (2) Since $H$ is finite, any injective map from $H$ to itself must be bijective.
explanation: The provided text is an excerpt from a mathematical article that discusses the relationship between two subgroup properties: being a finite subgroup and being a powering-invariant subgroup. The text includes a statement of the theorem and a detailed proof. The proof is structured with steps, each explaining a part of the reasoning.
  
  The text does not contain a question or an answer to a question. It is a formal mathematical exposition.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Finite implies powering-invariant
  
  This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., finite subgroup) must also satisfy the second subgroup property (i.e., powering-invariant subgroup)
  View all subgroup property implications | View all subgroup property non-implications
  
  ## Statement
  
  Suppose $G$ is a group and $H$ is a finite subgroup of $G$. Then, $H$ is a powering-invariant subgroup of $G$. Explicitly, if $p$ is a prime number such that $G$ is powered over $p$, then $H$ is also powered over $p$.
  
  ## Proof
  
  Given: A group $G$, a finite subgroup $H$, a prime number $p$ such that the map $x \mapsto x^p$ is bijective from $G$ to itself.
  
  To prove: The map $x \mapsto x^p$, restricted to $H$, is bijective from $H$ to itself.
  
  Proof:
  
  Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
  1 The map $x \mapsto x^p$ sends $H$ to within itself. $H$ is a subgroup.
  2 The map $x \mapsto x^p$ (restricted to $H$) is injective from $H$ to itself. The map $x \mapsto x^p$ is bijective from $G$ to itself. Step (1) The map is bijective on $G$, hence injective on $G$, hence its restriction to $H$ must also be injective.
  3 the map $x \mapsto x^p$ (restricted to $H$) is injective from $H$ to itself. $H$ is finite. Step (2) Since $H$ is finite, any injective map from $H$ to itself must be bijective.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an excerpt from a mathematical article that discusses the relationship between two subgroup properties: being a finite subgroup and being a powering-invariant subgroup. The text includes a statement of the theorem and a detailed proof. The proof is structured with steps, each explaining a part of the reasoning.
  
  The text does not contain a question or an answer to a question. It is a formal mathematical exposition.
  
  Label: no-question