# The Smallest Non-Cyclic Simple Group has Order 60

DevFeed: [The Smallest Non-Cyclic Simple Group has Order 60](<https://devfeed.tech/articles/the-smallest-non-cyclic-simple-group-has-order-60-40244.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2011/10/08/the-smallest-non-cyclic-simple-group-has-order-60/>)

Published: 2011-10-08T20:53:48Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>)

Tags: [classification](<https://devfeed.tech/tags/classification.md>), [group-theory](<https://devfeed.tech/tags/group-theory.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [primes](<https://devfeed.tech/tags/primes.md>), [simple-groups](<https://devfeed.tech/tags/simple-groups.md>), [theory](<https://devfeed.tech/tags/theory.md>)

## AI overview

A mathematical proof examines simple groups of order less than 60 and establishes that the smallest non-cyclic simple group has order 60. It uses group theory concepts including normal subgroups, group actions, and the Sylow theorems.

## Source excerpt

Preamble: This proof is not particularly elegant or insightful. However, it belongs in this gallery for two reasons. First, it is an example of the goal of most mathematics: to classify things. In the same way that all natural numbers can be built up from primes, every group can be built up from simple groups. So if we want to understand all groups, it suffices to understand the simple ones. Indeed, this project has been the collective goal of hundreds of mathematicians for the past hundred years, culminating in one awesomely gargantuan theorem: The Classification of Finite Simple Groups.