Commutator of a group is identity?

Kanchana
Messages
3
Reaction score
0
If the group G/[G,G] is abelian then how do we show that xyx^{-1}y^{-1}=1?

Thanx
 
Physics news on Phys.org
I'm not sure I understand your question. Is G or G/[G,G] that is abelian? Are x and y in G?
 
G/[G,G] is abelian and x,y are from G/[G,G]. Then [x,y]=1..?
 
G/[G,G] is ALWAYS abelian, no matter what G is, yet you are phrasing it as if it's an additional assumption. xyx^-1y^-1=1 is the same equation as xy = yx, just by multiplying on the right by y and then by x on both sides of the equation. To put it another way, x and y commute if and only if their commutator, xyx^-1y^-1 is equal to the identity.
 

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

A Near-Rings with Noncommutative Addition and Two-Sided Distributivity
Replies
4
Views
2K
I Dfns group action & group, written in terms of the other
Replies
26
Views
378
I Differences between left vs right actions in some group theory questions
Replies
1
Views
63
I Question about "A Group Epimorphism is Surjective"
Replies
13
Views
575
MHB Is a Group Abelian if Inverses Commute?
Replies
2
Views
1K
I Meaning of "passage to the quotient"?
Replies
3
Views
441
A Decomposition into irreps of compact Lie group
Replies
4
Views
52
How Is the Abelianization of a Lie Group Defined?
Replies
3
Views
2K
I Show that commutativity is a structural property
Replies
1
Views
1K
A Classification of reductive groups via root datum
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top