Abstract Nonsense

Crushing one theorem at a time

A Classification of Integers n for Which the Only Groups of Order n are Cyclic (Pt. II)


Point of Post: This is a continuation of this post.

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September 13, 2011 Posted by | Uncategorized | , , , , , , , , | Leave a comment

A Classification of Integers n for Which the Only Groups of Order n are Cyclic (Pt. I)


Point of Post: In this post we conglomerate and extend a few exercises in Dummit and Foote’s Abstract Algebra which will prove that the only positive integers n for which the only group (up to isomorphism) of order n  is \mathbb{Z}_n are integers of the form n=p_1\cdots p_m are distinct primes with p_i\not\equiv 1\text{ mod }p_j for any i,j\in[m].

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Motivation

This post will complete several lemmas/theorems which works towards proving not only that every group of order p_1\cdots p_m where p_i\not\equiv 1\text{ mod }p_j for any i,j\in[m] (greatly generalizing the statement that a group of pq for primes p<q with q\not\equiv 1\text{ mod }p is cyclic) but also that numbers of this form are the only numbers for which the converse is true (namely every group of order n is cyclic).

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September 13, 2011 Posted by | Uncategorized | , , , , , , , , , , | 1 Comment

   

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