Commutator of a group is identity?

Kanchana · May 25, 2014

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

Thanx

HallsofIvy · May 25, 2014

I'm not sure I understand your question. Is G or G/[G,G] that is abelian? Are x and y in G?

Kanchana · May 25, 2014

G/[G,G] is abelian and x,y are from G/[G,G]. Then [x,y]=1..?

homeomorphic · May 26, 2014

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.

Commutator of a group is identity?

Thread 'When are Markov Matrices also Martingales?'

Thread 'Derivation of equations of stress tensor transformation'

Similar threads

Hot Threads

I How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?

I Showing ##k[x_1,\ldots,x_n]/\mathfrak{a}## is finite dimensional

I How do we distinguish two different notations for cokernel and coimage?

I Localising a non integral domain at a prime

I ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?

Recent Insights

Insights Thinking Outside The Box Versus Knowing What’s In The Box

Insights Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

Insights Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

Insights What Exactly is Dirac’s Delta Function? - Insight

Insights Relativator (Circular Slide-Rule): Simulated with Desmos - Insight

Insights Fixing Things Which Can Go Wrong With Complex Numbers