What Is an Infinite Group with Exactly Two Elements of Order 4?

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SUMMARY

An infinite group with exactly two elements of order 4 can be constructed using the complex numbers, specifically through the multiples of the imaginary unit, i. The discussion highlights the group G defined by the multiplication modulo 5, where elements 2 and 3 both have an order of 4. This construction demonstrates the existence of such groups and provides a foundational understanding of group theory concepts related to order and structure.

PREREQUISITES
  • Understanding of group theory concepts, specifically group order
  • Familiarity with complex numbers and their properties
  • Knowledge of modular arithmetic, particularly multiplication modulo 5
  • Basic concepts of infinite groups and their characteristics
NEXT STEPS
  • Research the properties of infinite groups in abstract algebra
  • Study the structure of groups generated by complex numbers
  • Learn about group homomorphisms and their applications in group theory
  • Explore the concept of cyclic groups and their orders
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Mathematicians, students of abstract algebra, and anyone interested in the properties of infinite groups and their elements' orders.

tim656
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what is an infinite group that has exactly two elements with order 4?

i let G be an infinite group for all R_5 ( multiplication modulo 5) within this interval [1,7)
so i got |2|=|3|=4.

i'm not sure this is the right answer but i couldn't think of anything else at a moment. help please.
 
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tim656 said:
what is an infinite group that has exactly two elements with order 4?

i let G be an infinite group for all R_5 ( multiplication modulo 5) within this interval [1,7)
so i got |2|=|3|=4.

I'm not sure this is the right answer but i couldn't think of anything else at a moment. help please.
Hello tim656. Welcome to PF !

Think about some subset of the complex numbers.

What are the multiples of the imaginary unit, i ?
 

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