GROUP GRAPHER
Compute as group
Boxes
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Spacing
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Colors
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Groups
Enter "[expression] in [group]" to graph an expression in one of the following groups.
• \(\mathbb{Z}_n\) ("Zn"), the ring of integers mod \(n\).
For example, "(x - 32)^2 + (y - 32)^2 in Z128"
or
"x^2 - 51xy + y^2 in Z100".
• \(U(n)\) ("Un"), the corresponding group of units.
For example, "xy in U90".
• \(\mathcal{B}\) ("B"), the Boolean algebra with "0" as false and "1" as true.
Booleans can be combined with "+" and "*" for OR and AND, respectively, and with "'" or "-" for NOT.
For example, "x'y + xy' in B" produces a truth table for XOR.
"xy in B x B x B x B x B" produces a fractal.
• \(D_n\) ("Dn"), the group of symmetries of a regular \(n\)-gon in the plane.
Here a nonnegative number "m" represents a counterclockwise rotation by m, while a negative number "-m" represents "(m - 1)H", a horizontal flip followed by a counterclockwise rotation by m - 1.
For example, "xy in D80".
• \(\mathbb{F}_p[t]/(q), q(t) = c_0 + c_1 t + \cdots + c_nt^n\) ("Fp[c0, c1, ..., cn]"), the finite field of prime-power order \(p^n\) where \(q\) is irreducible over \(\mathbb{Z}_p\).
Polynomials can be entered as bracketed vectors, e.g., "xy + [1,1] in F3[1,0,1]" graphs \(xy + i + 1\) mod 3.
Other examples: "x + y in F7[1,1,1]"
and
"xy in F5[3,3,0,1]".
External direct products
To graph a Cartesian product, separate group names with "x", e.g., "xy in Z2 x Z4 x U5".
Operations
"+", "-", "*", "/" denote the usual operations, based on the algebraic setting.
Use "^" for exponents.
Use "'" for the inverse of a group operation, e.g., "xy' in U5" graphs \(xy^{-1}\) in \(U(5)\).
Equality checks
Using "=" to check for equality returns a Boolean.
Boolean clauses can be combined with the Boolean operations explained above.
For example, "y^2 = x^3 + x in Z23" graphs the solutions of \(y^2=x^3+x\) in the field \(\mathbb{Z}_{23}\).
"(x = y^3)(y = x^3) in Z10" plots \(x, y\) which are cubes of each other in \(\mathbb{Z}_{10}\).
Saving
Right click a graph to save it as an image.
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