@dharr CayleyTableGroup is another way. (CustomGroup could also be used.) - works for add and multiply

Download mod7.mw

I'm assuming you really want a group, rather than just doing arithmetic mod n. The addition case is just the CyclicGroup:

Download mod7.mw

Following @Mariusz Iwaniuk but can't get the last step.

Download summation.mw

I clicked on a pMML link (5.13.E1), which downloaded a file. I opened it in a text editor, selected all the text and copied it. In a Maple 2D input region, I pasted and it asked me if I wanted to convert MathML to 2D input, and I said yes. It looks a little strange in the input region (not as bad as here), but seems to be functionally OK.

Download PMML.mw

Perhaps you want to assume n is an integer, or even or odd, in which case the following may help. If you want to use the +/- symbol, then that is not as easily done.

Download Xn.mw

@Kitonum's method requires retyping of the variable names on the lhs. This can be avoided with assign.

Download solveEqs.mw

The ExportMatrix command can take a list of Matrices and write them to a MATLAB format file. See the help page for more details.

I don't know the exact answer. I found it confusing that you added the types in the module within the module; maybe it is a scoping issue. Why not just add them in the ModuleLoad procedure of RationalTrigonometry? (And for debugging purposes, I'd comment out any lines that remove them, in case they were added and the removed before you intended.)

Perhaps this is what you recall.

de:=gfun:-algeqtodiffeq(y = y^2*z + 1, y(z))

gives

The thing on the left-hand side of an assignment (:=) can't be an expression like lambda^m. You could assign to, say, lambda_m, but it would just be a name.

Perhaps you want something like eqn[m] instead of lambda^m?

Use mul for this, just

mul(v)

or

mul(i, i in v)

You do need to change to x=1/I to do linear least squares. 

Download Mapleprimes_Question_Book_2_Paragraph_5.12_Question_3.mw

Use

W := Matrix(2, 2, [[a, b], [c, d]])

not Array for "." to work correctly.

For varying initial conditions on one variable (x in the worksheet) use DEplot. Example in the attached worksheet.

Download DEplot.mw

des := {diff(x(t), t) = -x(t), diff(y(t), t) = -y(t)^2};
DEtools:-DEplot(des, {x(t), y(t)}, 0 .. 4,
                [[x(0) = 1, y(0) = 1],[x(0) = 2, y(0) = 1]], scene = [t, x]);

@tomleslie's generation of a CSV file is correct, but if you want it named as a .txt file just use

Export("E:\\M.txt", M, format = "CSV");

Your error message was because Export wants the filename before the data.