Dork Scratchings

Here's a puzzle: Multiply $142857$ by $2$ and you get $285714$ which has the same digits Multiply $142857$ by $3$ and you get $428571$ which has the same digits Multiply $142857$ by $4$ and you get $571428$ which has the same digits  In fact you keep getting permutations of $142857$  till you get to a multiple of $7$ at which point you get $999999$.   Why? I can't remember now how I figured it out, but I do remember not being able to sleep, then having some sort of eureka moment and running downstairs in the middle of the night to scribble down the answer. The solution involves some group theory, which I studied last century when I did my Maths degree at Bristol.  Periodically I have a little panic that I've forgotten everything I learned there, and I refresh my memory.  I think it must have been after one of these episodes that my subconscious linked it to this strange number that I probably first came across at school.  Anyway, the s...