Solutions for Herstein's Topics in Algebra - Introduction

Herstein’s Topics in Algebra is one of the well known abstract algebra text. It is super-clear in its writing style with its theorems introduced with enthusiastic motivations. Not too wordy, but not too compact that it misses important details. Although it is not an easy read, but any undergraduates drilling in math(or even beginning-graduates) will definitely benefit himself/herself by learning from this book. I bet that most of the modern-day abstract algebra books, ones like Fraleigh’s “First Course in Abstract Algebra” and Dummit & Foote’s “Abstract Algebra”, must have been influenced by this Herstein’s masterpiece.

The exercise problems from Herstein’s book are interesting and motivating at the same time. But a few problems lying are, there are some exercises(particularly, the ones with asterisks) which cannot be handled smoothly without introducing some additional concepts in relatively advanced algebra. So I have decided to give a challenge myself by making a (almost)complete solutions mannual for the exercises in the Herstein’s book. But I am just an undergradute, not even an extraordinary in my classroom. So there must be some flaws in my answers and writing skills(In fact, English is not my mother-tongue).

Here are some of the notes regarding my mathematical writing style:

1. I have tried to follow the Herstein’s method of notation. For an example, $o(G)$ for the order of a group $G$. Also, I kept using left multiplication notation for automorphism, that is, for an automorphism $T : G \to G $, $xT$ shall denote the mapped element. I tried to follow the meaning of ‘isomorphism’, which Herstein takes equivalent to ‘monomorphism’. But I tried to use a more set-theoritic notion, such like ‘bijective’ and ‘injective’, etc. I believe this would reduce confusions in writings.
2. Although Herstein did not explicitly introduced the notion of “Group action”, I shall make use of it unless there are some confusions or any chances of mis-interpretations in the contexts.
3. I am also a beginner in mathematics and especially in abstract algebra. There will be mistakes and missing details. I appreciate any type of comments regarded with my proofs and writings. Insightful criticisms are always welcome!

*Of course not every problems are solved by own(almost 90% are done by myself). References such from Lang and Rotman helped me lot. Stack Exchange was also one of my guides.

I will readily post a pdf containing more than one section, at least once in a week.