Group Theory, cyclic group proof

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SUMMARY

The discussion centers on proving that the group Z sub n is cyclic, specifically under the additive operation. The key conclusion is that Z sub n can be expressed as G = {nx; n exists in Z}, where x is chosen as 1, serving as the generator of the group. This proof highlights the necessity of using the additive operation rather than multiplicative, as the latter does not yield integers for all elements.

PREREQUISITES
  • Understanding of group theory concepts, specifically cyclic groups.
  • Familiarity with the notation and properties of integers, particularly Z sub n.
  • Knowledge of binary operations, specifically additive and multiplicative operations.
  • Ability to work with mathematical proofs and definitions in abstract algebra.
NEXT STEPS
  • Study the properties of cyclic groups in abstract algebra.
  • Learn about generators of groups and their significance in group theory.
  • Explore the differences between additive and multiplicative groups.
  • Investigate other types of groups, such as finite groups and their structures.
USEFUL FOR

Students of abstract algebra, mathematicians focusing on group theory, and educators teaching concepts related to cyclic groups and their properties.

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Homework Statement



Prove that Z sub n is cyclic. (I can't find the subscript, but it should be the set of all integers, subscript n.)

Homework Equations

Let (G,*) be a group. A group G is cyclic if there exists an element x in G such that G = {(x^n); n exists in Z.}

(Z is the set of all integers)

The Attempt at a Solution



* is a binary operation, and for my purposes, is either additive (+) or multiplicative (x).

Multiplicative does not work because the multiplicative inverse of, say, 2 is not an integer. So the operation must be additive. So I can rewrite the equation for (G,+) as:

G = {nx; n exists in Z}

but that's where I get stuck. Thanks for the help!
 
Last edited:
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How about taking x = 1 as your generator?
 
Every cyclic group has a generator.

What is your generator in this case?

edit: nm already beaten too it
 
thanks to both!
 

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