H ≅ gHg^(-1) prove that

Question

To prove it is isomorphism ?H ≅ gHg^(-1) prove it is isomorphic

Answer 1

I assume these are all matrices.

If H=gHg^-1 then let's multiply both sides by g, and we have:

Hg=gHg^-1g=(gH)(g^-1g),

but g^-1g=I the identity matrix so Hg=gH.

Similarly, multiplying by g^-1:

g^-1H=g^-1gHg^-1=(g^-1g)(Hg^-1)=Hg^-1.

Hence Hg=gH and g^-1H=Hg^-1.

We can also write g^-1Hg=H, that is, H=g^-1Hg, which makes gHg^-1=g^-1Hg.

If we replace gH on the left with Hg, and Hg on the right with gH, we simply get H=H so the there is isomorphism between the two constructs.