can a dog travel on south west trains with its able owner?

What's an able owner?

Otherwise, no. Pets aren't allowed in trains.

Yes they can but must be on lead and these days wearing a muzzle - seriously.

You can take your dog on the train free of charge. I went from Manchester to Cornwall last time, and travelled by train while I was there. So that would be on Southwest Trains, I should think, can't remember. You can go on all trains, I'm pretty sure.

No

Yes it can! My husbands a train driver and gave me this link for you. It has all the regulations for animals.
http://nationalrail.co.uk/passenger_services/luggage_animals.html

Yes. Pets can be taken on all National Rail services.

You may take your dog on any train in the UK (except, of course, Eurostar). Dogs must not occupy seats, nor are they permitted in sleeper compartments or restaurant cars.
However, the carriage of dogs on trains is at the discretion of the person in charge, i.e. the Train Manager, Senior Conductor, etc. If a dog was misbehaving. e.g. barking or causing alarm to passengers, it would be a case of "Walkies!" all the way to your destination.