[Math] problem book on Group theory after doing Fraleigh.

abstract-algebrabook-recommendationgroup-theoryreference-request

Please refer a problem book on

Group Theory:TOPICS: Groups, subgroups, Abelian groups, non-abelian groups, cyclic groups, permutation groups; Normal subgroups, Lagrange's Theorem for finite groups, group homomorphisms and basic concepts of quotient groups (only group theory).

I have already done A first course in abstract algebra by John b. Fraleigh. But I need one problem book to solve in order to be confident. I don't need a proof oriented problem book,my focus is to solve problems which are applications of theorems. I had Gallian on my mind, but it lacked solutions.

please provide your suggestions. thanks

Best Answer

The best problems on group theory for a beginner, to me, is still contained in the second chapter of I.N.Herstien's Topics in Algebra. This is the book I learned algebra from and it still has the best exercises of any textbook I've seen. This is the book you want,trust me.

Related Solutions

Abstract Algebra – Strategy for Counting Group Homomorphisms

In general, a group homomorphism is determined entirely by the image of a set of generators. Given this, if the codomain is finite, then this puts a relatively easy to check upper bound on the total number of possible homomorphisms. For example, in your second example question, $\mathbb{Z}_p\oplus\mathbb{Z}_p$ is generated by the elements $(0,1)$ and $(1,0)$ and so any homomorphism is entirely determined by where these elements are sent (and they must be sent to elements of order dividing $p$). I don't think you quite finished typing the entire question for question 2 because you didn't mention a codomain for the set of homomorphisms.

For your first example, note that $S_n$ can be given presentation $$\langle \sigma_i\ldots,\sigma_{n-1}|\:\:\sigma_i^2=1,$$ $$\sigma_i\sigma_j=\sigma_j\sigma_i\:\:\:\mbox{ for }j\neq i\pm 1,$$ $$\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}\rangle$$

Suppose $h\colon S_n\rightarrow G$ is a homomorphism. Then $h$ is determined by the image of the generators. But note, $$h(\sigma_i\sigma_{i+1}\sigma_i)=h(\sigma_i)h(\sigma_{i+1})h(\sigma_i)$$ $$=h(\sigma_i)h(\sigma_{i})h(\sigma_{i+1})=h(\sigma_i\sigma_{i})h(\sigma_{i+1})=h(1)h(\sigma_{i+1})=h(\sigma_{i+1})$$ and similarly $h(\sigma_{i+1}\sigma_{i}\sigma_{i+1})=h(\sigma_i)$. But $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ and so we have $h(\sigma_i)=h(\sigma_{i+1})$ for all $i$. By induction the image of all generators is equal and so $h$ is determined by the image of any single generator. Given that $h(\sigma_i)$ has to be an element of order dividing 2, this significantly reduces the possible homomorphisms.

[Math] the importance of examples in the study of group theory

There is some pedagogical value to understanding how theorems/definitions apply to and playout with particular groups (i.e., through examples). While it is admirable to yearn for a thorough theoretical understanding of group theory, try to think of studying and probing the examples as a means of testing your understanding of the theories and definitions, as they relate to specific groups. Being able to construct a proof is wonderful, but being able to use and apply your learning is crucial to obtaining a fluency with what you're learning. Being able to understand and apply the theorems and definitions is also a test of your comprehension, and will facilitate the proof-writing process as well. For example, being able to construct and anticipate counterexamples is crucial to proof-writing, and counterexamples will be hard to come by if you don't understand how particular groups exemplify or fail to exemplify group properties.

A great website is the website Groupprops - "A Group Properties Wiki". You'll find a menu to the left:

Terms/definitions
Facts/theorems
Survey articles
Specific information

and can quickly look-up (almost) anything/everything you'd want to know about (almost) any group.

Best Answer

Related Solutions

Abstract Algebra – Strategy for Counting Group Homomorphisms

[Math] the importance of examples in the study of group theory

Related Question