# Abstract Algebra: Dummit and Foote Exercise

• nateHI
In summary, in this problem, the question is asking for the smallest number m that satisfies ##\sigma^m = \text{id}## where ##\sigma## is a permutation written as a product of disjoint cycles. This is because the order of a cycle is its length, and the smallest m must be a multiple of the length of each cycle.
nateHI
This isn't homework, I'm just trying to refresh my memory on cyclic groups.

My question is, in this problem solution, how does ##{\sigma_i}^m=1## follow from ##\sigma_i## being disjoint?

It rather follows from ##\sigma_i## being a cycle.

It says that a permutation can be written as a product of disjoint cycle ##\sigma = \sigma_1 \circ ... \circ \sigma_p ##.
The question asks you to find the smallest ##m \ge 1## such that ##\sigma ^ m = \text{id}##.
Since the cycles are disjoint, you can commute all the cycles, and rewriting things nicely, ##m## has to be such that ##\sigma_i ^ m = \text{id} ##, for all ##i = 1 ... p ##.
Since the order of a cycle is its length, ##m## must be a multiple of the length of each cycle ##\sigma_i##. The smallest such ##m## is the least common multiple of all the lengths.

Got it thanks!

## 1. What is Abstract Algebra?

Abstract Algebra is a branch of mathematics that deals with the study of algebraic structures, such as groups, rings, and fields, and their properties. It focuses on abstract, general concepts rather than specific numbers or equations.

## 2. What is the purpose of Dummit and Foote's "Abstract Algebra" textbook?

The purpose of Dummit and Foote's "Abstract Algebra" textbook is to provide a comprehensive and rigorous introduction to the subject. It covers a wide range of topics and includes numerous exercises and examples to help readers understand the concepts and techniques of abstract algebra.

## 3. Who is the target audience for "Abstract Algebra" by Dummit and Foote?

The target audience for "Abstract Algebra" by Dummit and Foote is primarily advanced undergraduate and graduate students in mathematics. It can also be useful for researchers and professionals in related fields who want to deepen their understanding of abstract algebra.

## 4. How should I approach the exercises in "Abstract Algebra" by Dummit and Foote?

The exercises in "Abstract Algebra" by Dummit and Foote are designed to help readers strengthen their understanding of the material and develop problem-solving skills. It is recommended to read the corresponding sections in the textbook before attempting the exercises and to work through them systematically.

## 5. Are there any online resources available to supplement "Abstract Algebra" by Dummit and Foote?

Yes, there are various online resources available to supplement "Abstract Algebra" by Dummit and Foote, including lecture notes, additional exercises, and video lectures. However, it is important to use these resources as a supplement and not as a replacement for the textbook.

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