Group theory order calculator

Analyse the classic finite groups — cyclic Zₙ, direct products Zₘ×Zₙ, dihedral Dₙ and symmetric Sₙ. It gives the group order, whether it is abelian, its structure, the number of generators for cyclic groups, and the order of any element you enter, including a permutation in cycle notation for Sₙ.

ℤ/12ℤ → ord(8) = 3 — the tool shows every step in between, not just the final figure.

Anything modelled as points and connections: a shortest route, a network's capacity, a schedule with dependencies, or a circuit reduced to its logic.

Assuming a shortest path stays shortest when a weight changes sign. Negative edges break the greedy argument Dijkstra rests on, and the algorithm returns a confident wrong answer rather than an error.

They sit next to each other but answer different questions: Set theory calculator is the one to open when you need it to compute every core set operation at once from two, three or four sets: union, intersection, both differences, symmetric difference, complement, Cartesian product and power set. It also checks subset, equality and disjointness, and reports all the cardinalities — numbers, letters or words all work. Pick whichever matches what you're starting from — both are free.

First-Order ODE Solver is the closest one after this: Solves dy/dx = f(x, y) from an initial condition with RK4, integrating outward in both directions, and plots the solution curve. Growth, decay, Newton cooling, logistic and separable cases are one click away.

Order of Operations Calculator (PEMDAS) and Second-Order ODE Solver — they come up in the same task often enough to be worth a second tab.

The algorithms are the textbook ones and their results are exact for the graph you enter. What varies is cost: several of these problems have no known efficient solution, so large inputs are answered by heuristic and the tool says when that is the case.