# Galilean group transformations

My problem is the following: I have difficulties in answering questions (c), (d) and (e).

For (c) my answer was $\sqrt{x^{2}+t^{2}}$ and yes, the group forms the group of all isometries since the group contains those matrices that translate and rotate the 2D space.

For (d) I have said that there is no difference since E(2) is the symmetry group of 2D space.

For (e) I have tried using the theorem which states that if D is a irrep of a group G and there exists a matrix B which commutes with B then $B=\lambda 1$. Thus, I tried forming a 3x3 matrix and imposing conditions but that has only led me to the diagonal elements being equal and the elements below the main diagonal being zero. Is there some other theorem that I could apply?

Is this the way to approach the questions?

• Are you sure that Galilean boosts preserve the line element $\mathrm{d}x^2 + \mathrm{d}t^2$? – gj255 Apr 10 '17 at 16:14
• it certainly preserves the length $dx^2$ and the time interval $dt^2$ separately by ordinary geometry and Newtonian definition of time interval. – ZeroTheHero Apr 10 '17 at 21:47

• Part c) Surely the length is $x^2+t^2$.
• Part d) I don't think your matrix representation for $T(a,b,v)$ is $E(2)$. See in particular Eq.(1) of this paper.