MHB Show that the abelian groups are isomorphic

  • Thread starter Thread starter buckylomax
  • Start date Start date
  • Tags Tags
    Groups
buckylomax
Messages
3
Reaction score
0
Hi there,

I'm trying to figure out this question:

Let A=[aij] be a 3x3 matrix with integer entries and let B=[bij] be it’s transpose. Let P and Q be the Abelian groups represented by A and B respectively. Show that P and Q are isomorphic by comparing the effects of row and column operations on A and B.

I've very stuck with this question. I figure I need to reduce both matrices to diagonal form and then compare them but I'm not sure how to get there. Any advices would be appreciated.

Thanks

B.
 
Physics news on Phys.org
Can you be more precise about what you mean by "the Abelian groups represented by A and B"?
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Can an Abelian Group Be Isomorphic to a Non-Abelian Group in Physics?
Replies
6
Views
2K
MHB Is G/G isomorphic to the trivial group? A proof for G/G\cong \{e\}
Replies
1
Views
2K
I Show that commutativity is a structural property
Replies
1
Views
1K
How to find group types for a particular order?
Replies
2
Views
2K
MHB Is G an Abelian Group Given Specific Conditions?
Replies
5
Views
2K
MHB The automorphic group is abelian
Replies
14
Views
3K
MHB Direct sum of free abelian groups
Replies
1
Views
2K
MHB Showing that a group is non-abelian
Replies
7
Views
2K
MHB Proving Finite subgroups of the multiplicative group of a field are cyclic
Replies
3
Views
2K
MHB Prove Abelian Group: $(a \cdot b)^n = a^n \cdot b^n$
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top