Sec 8.2, 11/30/09

- It is interesting to know that all finite abelian group is a direct sum of cyclic subgroups and the orders of these cyclic subgroups are uniquely determined by the group.
- The Proof for Fundamental theorem of finite Abelian groups is little confusing, i did not understand the invariant factors, why it is two numbers in some cases and it is just one number as shown in one of the example in page 258 of the text book.

exam preparation, 11/17/09

- For better understanding before the exam one should know all the definitions and the theorems in each section of the chapters.
-can you go over the proof of third Isomorphism theorem for groups?

Sec 7.8,11/16/09

-this section follows from the topic quotient rings, basically its the same principle, unlike here the kernel is defined as those element of the domain that maps to the identity element of the image group in a homomorphism of groups. i also liked the definition of simple group i.e. and the group itself.
-i find difficulty in understanding the proof of First Isomorphic theorem for groups, other thing i am not totally clear is the subgroup of quotient groups. how is K/N a subgroup of G/N?

Sec 7.7,11/13/09

- It is interesting to know that if N is normal subgroup of a group G, then the group N, G and G/N are related and if we know enough information about two of these groups, we can determine useful information about the third.
- I didn't really understand when it says that G/N denotes the set of all right cosets of N in G, is it true that it is also the set of all left cosets of N in G?

Sec 7.6, 11/11/09

-It is interesting to know that properties of left congruence and right congruence has the same basic properties. and more over the condition Na=aN does not imply an=na for n in N.
-I did not understand why (inverse a)Na=N does not mean that ( inverse a)na for n in N why it is that ( inverse a)na=n1 for n1 in N, why can't it be just n? Is it just a different symbol or it has some meaning?

Sec 7.5, 11/09/09

- The section is fairly easy and straight forward, it interesting to know that every group of order p is cyclic and isomorphic to Zp.
- I have still some problem in understanding the operation table in page 205.

Sec 7.5, 11/6/09

-the most interesting part me in this section is that the property of congruence still holds within the element of a group.
-i did understand what the Lagrange theorem is saying but i did not understand the proof for the theorem.