There seems to be a lot of confusion as what constitutes the glis glis both in this forum and from my internet browsing in the past hour or so. I can be included in the ranks of the confused.
I may however have managed to answer my own question earlier as to whether the glis glis is a protected species in France or not.
Linnaeus who categorised species generally in 1766 seems to have put the little creature in one family of mammals but somehow over the centuries its status has changed. Ok I'm getting nerdy I know and I'm no biologist. It's only important because all animals in the family GLIRIDAE are on the current protected species list and to quote from www.thepiedpiper.co.uk in an article on Myoxus Glis (Glis Glis)
'The taxonomy of this creature has changed. Many references refer to MYOXUS GLIS or Glis Glis and place it in the family GLIRIDAE rather than MYOXIDAE'
So if it's NOT a member of the family Gliridae it's not protected it would seem.
Confusion still reigns however in many minds it would seem as MYOXUS GLIS is apparently also known as MYOLUX GLIS together with common names of Glis Glis or Loir Gris. All refer to the same furry mammal.
The garden dormouse is ELIOMYS QUECINUS and known in France, as David Gay has already pointed out, as the lerot commun or loir des greniers. Just to add even more confusion I also read that 'the garden dormouse or Muscardinus Avellanarius is the only dormouse listed as a protected species.'
All of which is very interesting but doesn't help Martha to get rid of the pests she has, whatever name they go by. It would seem they can be legally zapped here in France though.