# Abstract Nonsense

## 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.

September 13, 2011

## 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 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).

September 13, 2011