Finite abelian group textbook help

In summary, the conversation discusses the search for literature on finite abelian groups, specifically on the action of $\text{Aut}(A)$ on $A$. The speaker has searched on Amazon and Google but has not found a dedicated book on the topic. They mention that finite abelian groups are well characterized and suggest a potential source of literature.
  • #1
caffeinemachine
Gold Member
MHB
816
15
I need to read about finite abelian groups.

I searched 'finite abelian group' on amazon and the closest search result was 'finite group theory'. Googling didn't help either.

Does there exist a book dedicated to finite abelian groups? If yes, and if you know of a good one then please reply.

Right now I am primarily concerned with the action of $\text{Aut}(A)$ on $A$ where $A$ is a finite abelian group. If you know of a good source of literature in this particular topic then please help.
 
Physics news on Phys.org
  • #2
I doubt there will be books written exclusively on finite abelian groups, because finite abelian groups are quite well characterized by the results here. Also, I have not read it, but this might be of interest.
 

FAQ: Finite abelian group textbook help

What is a finite abelian group?

A finite abelian group is a mathematical structure consisting of a finite set of elements and an operation that combines two elements to produce a third element. The operation is commutative, meaning the order in which the elements are combined does not affect the result. Additionally, every element in the group has an inverse, meaning there is another element that, when combined with it, produces the identity element (an element that does not change the value of other elements when combined with them).

What are some examples of finite abelian groups?

Common examples of finite abelian groups include the integers under addition, the integers modulo n under addition (also known as cyclic groups), and the direct product of two or more cyclic groups. Other examples can be constructed using modular arithmetic or by taking the direct product of other finite groups.

What is the significance of studying finite abelian groups?

Finite abelian groups have many applications in mathematics, physics, and computer science. They provide a foundation for understanding more complex algebraic structures, and their properties can be used to prove theorems and solve problems in various fields. In particular, they are important in number theory, cryptography, and group theory.

How do I determine the order of a finite abelian group?

The order of a finite abelian group is simply the number of elements in the group. If the group is cyclic, the order can be determined by finding the smallest positive integer that, when multiplied by each element in the group, produces the identity element. For non-cyclic groups, the order can be found by multiplying the orders of each of the group's cyclic subgroups.

Are there any helpful resources for learning about finite abelian groups?

Yes, there are many textbooks and online resources available for learning about finite abelian groups. Some popular textbooks include "Algebra: Chapter 0" by Paolo Aluffi, "A First Course in Abstract Algebra" by John B. Fraleigh, and "Groups, Rings, and Fields" by J.A. Gallian. Additionally, there are many online lectures, videos, and practice problems available on websites such as Khan Academy and Math StackExchange.

Similar threads

I Decomposition per the Fundamental Theorem of Finite Abelian Groups
Replies
13
Views
2K
MHB Tensor Products of Modules and Free Abelian Groups based on Cartesian Product
Replies
2
Views
2K
MHB Free Abelian Groups .... Aluffi Proposition 5.6
Replies
4
Views
1K
MHB Irreducible representations of a slightly non-abelian group.
Replies
1
Views
2K
Group Theory: Finite Abelian Groups - An element of order
Replies
3
Views
2K
MHB Direct product of abelian groups. Isomorphism.
Replies
4
Views
2K
Finite Abelian Group: Showing Isomorphism & Subgroups of Order/Index p
Replies
1
Views
5K
MHB Tensor Products - D&F - Extension of the scalars
Replies
2
Views
1K
Finite non-abelian group of the order pq, pq primes
Replies
2
Views
5K
Understanding abelian Galois groups
Replies
16
Views
8K

Hot Threads

Recent Insights

Back
Top