Hi Marco and all,
I had Darij's problem as well, and many others probably did as well.
> In a right action, I would prefer p(1) to give a warning. In a right
> action, I would want some notation where p is on the right, preferably
> 1^p (1 hat p).
>
That would make sense (except that I don't really see why "^" is better
than "*", see below). In principle one can even allow completely symmetric
notation:
- left action of g on x: g(x) or g^x; think of [left exponent g]x in
two-dimensional notation
- right action of g on x: (x)g or x^g
Of course g^x and (x)g look a bit funny and maybe too confusing, but this
is just because we are used to thinking that g^x means that x is in the
exponent (as opposed to g, on the left), and we are not used to (x)g at
all. I guess existing parsers could be enhanced to accept all these
notations if somebody is crazy enough to want them. 8-)
The notation "*" has the wrong distributive laws in case of actions on
> rings or groups. Of course this is irrelevant for permutations acting
> on sets, but since Galois groups can be interpreted as permutation
> groups too and they act on rings, the hat is much better.
>
For both left and right actions, whether multiplicative ("*", similar
binary symbols or the empty notation) or exponential notation ("^", left or
right exponents) looks more natural depends on whether you are looking at
the behaviour of the group action with respect to addition or with respect
to multiplication. The following (and their equivalents for right actions)
look OK:
g*(x + y) = g*x + g*y
[left exponent g](x*y) = [left exponent g]x * [left exponent g]y
g^(x*y) = (g^x)*(g^y) (as long as you think of g as the exponent, not x
and y)
But the following look somewhat less appropriate:
g*(x*y) = (g*x)*(g*y)
[left exponent g](x + y) = [left exponent g]x + [left exponent g]y
(especially strange for right actions)
g^(x + y) = g^x + g^y
Peter
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