Abstract Algebra book with exercise solutions recommendations.

A simple search on Google shows the following:

• Abstract Algebra Manual ( Problems and Solutions ) by Ayman Badawi

• http://www.math.niu.edu/~beachy/abstract_algebra/review/review.pdf

• http://shell.cas.usf.edu/~wclark/Elem_abs_alg.pdf

• https://www.dpmms.cam.ac.uk/~oh233/old/algebra1/book.pdf

• Schaum's Theory & Problems of Abstract Algebra by Joong Fang

Gallian's book has selected exercises' solutions (apparently all the odd numbered ones)

Weiss's "First Course in Algebra and Number Theory" has lots of solutions

Whitesitt's "Principles of Modern Algebra" has many solutions

Krechmar's "A Problem Book in Algebra" (high school level) has all the solutions

Vinberg's "A Course in Algebra" has some solutions

Faddeev-Sominskii's "Problems in Higher Algebra" (high school-basic linear algebra) is all solutions

Gilbert's "Elements of Modern Algebra" has many solutions.

Etc.

I have read contemporary abstract algebra by Gallian. Though I dont remember if it had exercises and solutions but the chapters of the book are written in a very easy to understand way. It gave me a good insight into group theory.

I dont know if it's the answer to your question but you can check the book out if it works for you.