I am reading Kane - Reflection Groups and Invariant Theory and need help with property A-4 of the properties of reflections stated on page 7

(see attachment - Kane _ Reflection Groups and Invariant Theory - pages 6-7)

On page 6 Kane mentions he is working in dimensional Euclidean space ie where has the usual inner product (x,y).

In defining reflections with respect to vectors Kane writes:

" Given let be the hyperplane

We then define the reflection by the rules

if

"

Then Kane states that the following property follows:

(A-4) If $\displaystyle \phi $ is an orthogonal automorphism of E then

$\displaystyle \phi \cdot H_{\alpha} = H_{\phi \cdot \alpha} $

$\displaystyle \phi s_{\alpha} {\phi}^{-1} = s_{\phi \cdot \alpha $

Can someone explicitly and formally demonstrate A-4?

I am not really sure of the meaning of $\displaystyle H_{\phi \cdot \alpha} $ but I assume

$\displaystyle H_{\phi \cdot \alpha} = \{ x \ | \ (x, {\phi \cdot \alpha ) = 0 \}$

And I assume that

$\displaystyle \phi \cdot H_{\alpha} = \{ \phi \cdot x \ | \ (x, \alpha \} = 0 \}$

Can anyone help me formulate an explicit and detailed proof of A-4

Peter

Peter