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.
The text passage contains a mathematical question and its answer. The question is about proving that the intersection of two subgroups H and K of a group G, where the orders of H and K are relatively prime, is the trivial subgroup {1}. The answer is provided in the form of a proof that any element in the intersection must have an order of 1, implying that the intersection is {1}.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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.
user
The text passage contains a mathematical question and its answer. The question is about proving that the intersection of two subgroups H and K of a group G, where the orders of H and K are relatively prime, is the trivial subgroup {1}. The answer is provided in the form of a proof that any element in the intersection must have an order of 1, implying that the intersection is {1}.

TYPE: SINGLE_QUESTION_WITH_ANSWER