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What is the smallest positive integer n such that there are exactly 3 nonisomorphic Abelian group of order n
Challenge 25: Finite Abelian Groups
In summary, the smallest positive integer n for which there are exactly 3 nonisomorphic Abelian groups of order n is 8. This is determined by the fact that n cannot be square-free and the first two numbers with this property are 4 and 8, leading to 2 and 3 nonisomorphic groups respectively. Despite a small misunderstanding, the use of searching skills helped in reaching the correct answer.
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ShayanJ
Gold Member
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Its 8!
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Show your work! :)Shyan said:
Last edited:
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ShayanJ
Gold Member
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I thought its legitimate to use our searching skills!:DGreg Bernhardt said:
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mfb
Mentor
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n cannot be square-free (it needs factors that are multiples of each other), otherwise you don't get multiple non-isomorphic groups. The first two numbers with that property are 4 (leading to 2 different groups, corresponding to "4" and "2x2") and 8 ("8", "4x2", "2x2x2"). Therefore, 8 is the smallest n.
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DarthMatter
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Shyan said:
Oh come on!
If that's not a big spoiler, I don't know what is.
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Joffan
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Haha, I misread the challenge as asking for 3 non-Abelian groups, so - also using searching skills - I came to a different answer.
Related to Challenge 25: Finite Abelian Groups
1. What is the concept of Challenge 25: Finite Abelian Groups?
Challenge 25: Finite Abelian Groups is a mathematical problem that involves finding a group of integers that satisfy specific conditions, such as being finite and having an abelian (commutative) operation.
2. What is the significance of this challenge in mathematics?
This challenge is important in mathematics because it helps to develop problem-solving skills and deepen understanding of abstract concepts like group theory. It also has practical applications in cryptography and coding theory.
3. What are some strategies for solving Challenge 25: Finite Abelian Groups?
One strategy is to use the Fundamental Theorem of Abelian Groups, which states that any finite abelian group is isomorphic to a direct product of cyclic groups. Another strategy is to use the Chinese Remainder Theorem to break down the problem into smaller, simpler cases.
4. Are there any real-life examples of finite abelian groups?
Yes, there are many real-life examples of finite abelian groups, such as the set of rotations of a regular polygon, the set of symmetries of a cube, and the set of hours on a clock. These groups exhibit the properties of closure, associativity, identity, and inverse, making them ideal for modeling various phenomena in the physical world.
5. What are some other interesting problems related to finite abelian groups?
There are many interesting problems related to finite abelian groups, such as the classification of all finite abelian groups, finding the number of subgroups of a given order, and determining the automorphism group of a given abelian group. These problems have practical applications in fields such as chemistry, physics, and computer science.
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