text: 1. ## Relatively-prime-ordered subgroups
  
  If H and K are subgroups of a group G and if |H| and |K| are relatively prime, prove that $\displaystyle H \cap K = \{1\}$.
  
  I realize that this amounts to proving that $\displaystyle |H \cap K| = 1$, since the intersection of any family of subgroups is a subgroup. However, I don't know how where to go from here. Any help would be greatly appreciated!
  
  2. Originally Posted by jstew
  If H and K are subgroups of a group G and if |H| and |K| are relatively prime, prove that $\displaystyle H \cap K = \{1\}$.
  
  I realize that this amounts to proving that $\displaystyle |H \cap K| = 1$, since the intersection of any family of subgroups is a subgroup. However, I don't know how where to go from here. Any help would be greatly appreciated!
  Let $\displaystyle a\in H\cap K$ then the order of $\displaystyle a$ divides both $\displaystyle |H|$ and $\displaystyle |K|$. This forces the order to be one. Thus, what does it mean?
  
  3. Ah, so |a|=1 which implies that a={1}. But a was chosen arbitrarily, so {1} is the only element in $\displaystyle H \cap K$. Is this correct?
  
  4. Originally Posted by jstew
  Ah, so |a|=1 which implies that a={1}. But a was chosen arbitrarily, so {1} is the only element in $\displaystyle H \cap K$. Is this correct?
  Yes
  
  5. Thank you for the quick, helpful replies! You rock!
  
  6. Originally Posted by jstew
  You rock!
  I know. When I was a younger kid my mother used to tell me that I am the greatest person in the world. That I know everything. And that I am the most handsome.
explanation: The document contains a math question and a series of responses that provide an answer to the question. The question is about proving that the intersection of two subgroups with relatively prime orders is the trivial subgroup containing only the identity element.
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.
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distilabel_metadata.role: system
distilabel_metadata.content: 1. ## Relatively-prime-ordered subgroups
  
  If H and K are subgroups of a group G and if |H| and |K| are relatively prime, prove that $\displaystyle H \cap K = \{1\}$.
  
  I realize that this amounts to proving that $\displaystyle |H \cap K| = 1$, since the intersection of any family of subgroups is a subgroup. However, I don't know how where to go from here. Any help would be greatly appreciated!
  
  2. Originally Posted by jstew
  If H and K are subgroups of a group G and if |H| and |K| are relatively prime, prove that $\displaystyle H \cap K = \{1\}$.
  
  I realize that this amounts to proving that $\displaystyle |H \cap K| = 1$, since the intersection of any family of subgroups is a subgroup. However, I don't know how where to go from here. Any help would be greatly appreciated!
  Let $\displaystyle a\in H\cap K$ then the order of $\displaystyle a$ divides both $\displaystyle |H|$ and $\displaystyle |K|$. This forces the order to be one. Thus, what does it mean?
  
  3. Ah, so |a|=1 which implies that a={1}. But a was chosen arbitrarily, so {1} is the only element in $\displaystyle H \cap K$. Is this correct?
  
  4. Originally Posted by jstew
  Ah, so |a|=1 which implies that a={1}. But a was chosen arbitrarily, so {1} is the only element in $\displaystyle H \cap K$. Is this correct?
  Yes
  
  5. Thank you for the quick, helpful replies! You rock!
  
  6. Originally Posted by jstew
  You rock!
  I know. When I was a younger kid my mother used to tell me that I am the greatest person in the world. That I know everything. And that I am the most handsome.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a series of responses that provide an answer to the question. The question is about proving that the intersection of two subgroups with relatively prime orders is the trivial subgroup containing only the identity element.
  
  Label: question-and-answer