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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