Hello, this is a discussion page for the college students who are in various prestigious colleges throughout India, and are keen to pursue Mathematics.
Abstract Algebra plays a pivotal role in college mathematics, and it mainly focuses on three things
GROUPS, RINGS, and FIELDS.
Though Field is not in the course of some colleges, eventually it will be very helpful.
Now this discussion will be on some of the popular topics from Groups, Rings and Fields(to some extent).
AT A GLANCE WHAT IS IMPORTANT IN GROUPS
I am going to discuss chapterwise so in this post I give the first chapter of Group (Taking Herstein as reference)
Topics to study, and Problems that helps to build concepts (Chapter 2)
Definition of a group (Abelian, Cyclic imp) I.N. Herstein (Sec 2.3 prob 4,8,11 are imp for starters)
Hint of the problems
4) (Sol hint) Just take three consecutive integers like k, k+1, k+2 and try to change
\[ab = a^kab/a^k\], from where you will actually get \[ba^k=a^kb\]
Similarly try to find the same relation using k+2. you will get it.
8) (Sol hint) The group is finite this is your hint and use the closure property of the group. to give you a little more hint closure always doesn’t happen like ab it is also a.a and a.a.a.a like this. Try to use that.
11) (Sol hint)
This problems tests you on the “uniqueness of the inverse”, property of a group. Remember the group is of even order containing identity as an element. So without it there are odd no of elements in that group and all the elements has a unique inverse. So what now!! DO IT YOURSELF!!
SOME IMPORTANT THEOREMS IN THIS CHAPTER (I am taking Herstein as my reference Chap 2)
Uniqueness and existence of inverse and identity in a group (With proof). (Always watch the operation closely)
Very carefully understand the left and right multiplication. (One of the reasons why group theory become more absurd sometimes)
If you become comfortable then try problems like 14, 18, 19 (Chap 2)
14) (Sol hint) You are already given that your operation is product, the finite set you have abides associativity and closure property of a group.
Now use the cancellation property to establish the identity and inverse property.
First try using the fact that the group is finite then use the closure property to show the existence of identity. (Problem 8 will be helpful) From there using cancellation you can get the inverse property also.
18) (Sol Hint) Don’t use the given hint, you now know something about groups and matrices too, so why not put the noncommutative criteria of matrices under multiplication in good use??
19) You now can solve it can’t you?? Go on then. Good Luck.
As you are all starting as reference you can use books like:
1) Contemporary Abstract Algebra (J. Gallian Cengage)
2) Abstract Algebra (Dummit Foote)
3) Topics in Algebra (I.N. Herstein)
4)Higher Algebra by S.K.Mapa (This is a very basic book if you are not comfortable with the previous mentioned books then build your basic from this one)