Showing the unit group is cyclic

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Homework Help Overview

The discussion revolves around the unit group Up of the integers modulo a prime p, specifically exploring whether Up is cyclic and its structure in relation to Z/Z(p − 1).

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the definition of the unit group and its elements, questioning what constitutes invertible elements in Z/Zp. There is also confusion about the nature of the group and its elements, particularly regarding the inclusion of zero.

Discussion Status

Some participants have clarified the definition of the unit group and acknowledged the number of elements it contains. However, there remains uncertainty about the implications of these definitions and the structure of the group.

Contextual Notes

Participants are navigating the definitions and properties of the unit group, with specific attention to the role of zero and the total count of invertible elements.

Locoism
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Homework Statement


Let p be a positive prime and let Up be the unit group of Z/Zp. Show that Up is
cyclic and thus Up \cong Z/Z(p − 1).


The Attempt at a Solution


What do they mean by the unit group? Is that just the identity? Is it the group [p]? I'm lost without starting the question...
 
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The unit group is just the group of all invertible elements (with multiplication as operation). So x\in U_p if and only if x is invertible in \mathbb{Z}_p. You have to show that it is cyclic (generated by 1 element).
 
OK thank you!
 
But now that I think of it, wouldn't Up be {e,...,p}, the entire equivalence class of Zp, since every element can be multiplied by another (its respective inverse) to get the identity?
 
Locoism said:
But now that I think of it, wouldn't Up be {e,...,p}, the entire equivalence class of Zp, since every element can be multiplied by another (its respective inverse) to get the identity?

0 doesn't have an inverse...
 
oh right, so then U would have p-1 elements. Got it!
 

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