What are the elements of each order in D_n+Z_9 for n = 7 and 11?

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SUMMARY

The discussion focuses on determining the elements of each order in the group Dn + Z9 for n = 7 and 11, where both numbers are products of distinct primes. The participants clarify that D7 + Z9 and D11 + Z9 must be analyzed separately. The key conclusion is that the total number of elements of each order must equal the order of the group, which is essential for verifying the calculations.

PREREQUISITES
  • Understanding of group theory, specifically dihedral groups and cyclic groups.
  • Familiarity with the notation Dn and Zn.
  • Knowledge of prime factorization and its implications in group orders.
  • Ability to perform calculations involving group orders and element orders.
NEXT STEPS
  • Study the structure of dihedral groups, particularly Dn for n = 7 and 11.
  • Learn about the properties of cyclic groups, specifically Z9.
  • Research how to compute the order of elements in direct sums of groups.
  • Explore examples of verifying group orders and element distributions in similar group structures.
USEFUL FOR

This discussion is beneficial for students and researchers in abstract algebra, particularly those studying group theory, as well as educators teaching these concepts in advanced mathematics courses.

ccrfan44
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Pick a number n which is the product of 2 distinct primes 5 or more. Find the number of elements of each order in the groupd D(sub)n+Z(sub)9, completely explaining your work. Verify that these number add up to the order of the group.

Ive used 7 and 11 as my primes. So now do I use these primes in D_n since to where i get D_7+Z_9 and D_11+Z_9? This is where I am confused.
 
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Also once i have that info, then where do i go from here?
 

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