problem book on Group theory after doing Fraleigh. 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
 A: 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. 
A: The book Contemporary Abstract Algebra by Joseph Gallian contains large number of problems with increasing order of difficulty. 
For some advanced problems, one may refer to two problem books in group theory:


*

*Problems in Group Theory - Dixon

*Group Theory: Selected problems - B. Sury.
Another book on Group Theory with selected problems is "Groups and Representations: Alperin and Bell". It contains nice selected problems. However, the author(s) of the book say in the preface that the problems are unordered according to difficulty, with some philosophical thought. 
A: Have a look into Problems in Group Theory - John Dixon (Dover ed.).
A: Algebra 3rd edition by Mac Lane and Birkhoff.  
A: You can also take a look into 'Algebra: a graduate course' and 'Finite group theory' both by I Martin Isaacs. Both are amazing books, I read the theory from the first book and solved problems from both books.
A: *

*Dummit and Foote

*Rotman

These two have a very rich collection of problems along with theory.
Along with this,


*Problems in Group Theory by B.Sury

is an excellent problem book.
A: As for the problems, I wouldn't forget the not yet mentioned Herstein's Abstract algebra, possibly even more complete (from this standpoint) than his other, more popular Topics in algebra.
