1. ## Langrange theorem problem....

Find [ Z_40 : <[12], [20]> ]?

I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.

Thanks for the help!

2. ## Re: Langrange theorem problem....

Originally Posted by vedicmath
Find [ Z_40 : <[12], [20]> ]?

I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.

Thanks for the help!
So, it is "well known" that any subgroup of a cyclic group is cyclic. As you have pointed out, gcd(12, 20)=4. This means that the subgroup generated by 12 and 20 is equal to the subgroup generated by 4, $\displaystyle \langle 12, 20\rangle=\langle 4\rangle$ (this should be in your notes, and is the crux of the argument - if you do not know it, learn it!). So, what is the index of $\displaystyle \langle 4\rangle$ in $\displaystyle \mathbb{Z}_4$? Divide 40 by this order to get your answer (why?).
The text passage contains a mathematical question related to group theory, specifically about finding the index of a subgroup in a cyclic group. The response provides a detailed explanation and a step-by-step approach to solving the problem, but does not explicitly state the final answer.
single_question_with_answer

however, the answer is implied in the steps provided, so it can be considered as a single_question_with_answer. the final step is to divide 40 by the order of the subgroup generated by 4, which is 10, resulting in an index of 4.
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. ## Langrange theorem problem....

Find [ Z_40 : <[12], [20]> ]?

I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.

Thanks for the help!

2. ## Re: Langrange theorem problem....

Originally Posted by vedicmath
Find [ Z_40 : <[12], [20]> ]?

I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.

Thanks for the help!
So, it is "well known" that any subgroup of a cyclic group is cyclic. As you have pointed out, gcd(12, 20)=4. This means that the subgroup generated by 12 and 20 is equal to the subgroup generated by 4, $\displaystyle \langle 12, 20\rangle=\langle 4\rangle$ (this should be in your notes, and is the crux of the argument - if you do not know it, learn it!). So, what is the index of $\displaystyle \langle 4\rangle$ in $\displaystyle \mathbb{Z}_4$? Divide 40 by this order to get your answer (why?).
user
The text passage contains a mathematical question related to group theory, specifically about finding the index of a subgroup in a cyclic group. The response provides a detailed explanation and a step-by-step approach to solving the problem, but does not explicitly state the final answer.

TYPE: SINGLE_QUESTION_WITH_ANSWER

However, the answer is implied in the steps provided, so it can be considered as a SINGLE_QUESTION_WITH_ANSWER. The final step is to divide 40 by the order of the subgroup generated by 4, which is 10, resulting in an index of 4.