Group Elements of Z24: Find the Order

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SUMMARY

The discussion focuses on determining the order of every element in the additive group Z24, which consists of integers modulo 24. Participants clarify that Z24 has 23 non-zero members, and the order of an element 'a' is defined as the smallest integer 'n' such that na = 0 (mod 24). For instance, the element 6 has an order of 4 because 6 added to itself four times equals 24, which is congruent to 0 modulo 24. Conversely, elements like 5, which are not divisors of 24, have an order of infinity.

PREREQUISITES
  • Understanding of modular arithmetic, specifically additive groups.
  • Familiarity with the concept of the order of an element in group theory.
  • Basic knowledge of integer properties and divisibility.
  • Experience with mathematical notation and operations involving congruences.
NEXT STEPS
  • Study the properties of additive groups in modular arithmetic.
  • Learn how to calculate the order of elements in different groups.
  • Explore the relationship between divisors and element orders in modular systems.
  • Investigate other groups, such as Z12 or Z30, to compare element orders.
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Mathematicians, students studying group theory, and anyone interested in modular arithmetic and its applications in abstract algebra.

Yara Leonard
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Determine the order of every element of Z24
 
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kindly infrom what you have tried and where you are facing problems
 
Do you understand that "z24" is the additive group of integers modulo 24? That has 23 non-zero members so you will need to give 23 answers. Do you further understand that the "order" of a member, a, of a group is the integer "n" such that na= 0 where "na" means a added to itself n times. For example 6+ 6+ 6+ 6= 24= 0 (mod 24) so 6 has order 4. No multiple of 5 will be 24 so the order of 5 (and any number that is not a divisor of 24) is "infinity".
 

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