🕹️ Practice arcade

🔄 Element Order Hunt

Given an element of Z_n, find its order and check whether it generates the whole group -- all from one gcd computation.

🧮 Totient Sprint

Compute Euler's totient function as many times as you can before the clock runs out.

🔀 Permutation Structure Drill

Given a permutation's cycle type, find its order and its parity -- lcm, not sum; transposition count, not cycle count.

📐 Subgroup Structure Sprint

Lagrange's theorem and direct-product element orders -- index from two orders, and lcm (not sum) for a paired element's own order.

🎯 Sylow Subgroup Sprint

Factor |G| and pick out a Sylow p-subgroup's exact order -- the full power of p, never just p, never the whole group.

⊗ Direct Product Structure Sprint

Multiply two orders to get |G x H|, then check gcd(m,n) to see whether Z_m x Z_n is cyclic at all.

🔢 Group Order Formulas Sprint

|D_n|=2n, |S_n|=n!, |A_n|=n!/2 -- three families, three formulas, one quick drill.

🔁 Cyclic Group Structure Sprint

How many subgroups a cyclic group has, and how many of its own elements actually generate it -- one divisor count, one totient.

🎡 Orbit-Stabilizer & Burnside Sprint

Solve |orbit|*|stabilizer|=|G| for the missing piece, then average fixed points across a whole group to count orbits.

🧬 Quotient & First Isomorphism Sprint

|G/N| and |im(phi)| are the same division wearing two different theorems -- divide by the smaller order, never multiply.

🧮 Onto, Bijection & Composition Sprint

Decide whether a function is onto, whether it's a full bijection, or compose two explicit functions at a point.

🧮 Kernel & Injectivity Sprint

Divide |G| by |im(phi)| for |ker(phi)|, then read injectivity straight off the kernel's own size -- trivial kernel or nothing.

🔄 Automorphism Counting Sprint

Count Aut(Z_n) with a totient, or Inn(G) with a center-quotient -- both a division, never a subtraction.

🧬 Second & Third Isomorphism Theorem Sprint

|HN|=|H||N|/|H∩N| for the second theorem, |G|/|H| (not |G|/|K|) for the third's double quotient.

📐 Index & Normalizer Sprint

Index 2 is always normal, no test needed -- and a normalizer's order is |G| divided by the conjugate count, never the count itself.

🔺 Dihedral Invariants Sprint

|Z(D_n)| and |D_n^ab| both split on the parity of n -- the same half-turn fact driving two different quotients.

🌳 Sylow Counting Sprint

Find the one forced value of n_p from the cofactor's own divisors, then read normality straight off whether n_p=1.

🪜 Composition Series Sprint

Count Z_n's composition-series length by prime factors with multiplicity, then check whether Jordan-Hölder forces identical chains (it doesn't).

🏛️ Classification Depth Sprint

Multiply partition counts across two distinct primes at once, and confirm order p^2 never has a non-abelian case.

🔐 Euler's Theorem Survival

Reduce the exponent mod phi(n) before computing a^k mod n -- one bad reduction ends the round.

🪞 Isomorphism Recognition Survival

Z_m x Z_n vs Z_mn -- matching order alone never proves isomorphic, only gcd(m,n)=1 does. One wrong verdict ends the round.

⚖️ Conjugacy & Normality Verdicts Survival

Is a conjugacy class a subgroup? Does normality fix every element? Two setwise-vs-elementwise traps, one wrong call ends the round.

🎯 p-Group & Sylow Existence Survival

A nontrivial p-group's center is never trivial, and Sylow's first theorem's existence guarantee never implies uniqueness on its own.

🧩 Structure Theory Verdicts Survival

Simple isn't small, solvable isn't abelian, and nilpotent isn't just solvable-with-a-different-name -- one wrong verdict ends the round.

🏛️ Classification Count Mastery

How many groups of a given order exist up to isomorphism -- a partition count for p^k, a divisibility test for pq. Five clean answers in a row, no clock.

🎯 Proof Technique Mastery

Direct, contrapositive, or contradiction -- name what each technique assumes at the start, five correct in a row.