Revision #673 → #1610 · back to history
addedSymmetric group (lead)c901d2919078
addedOrder of symmetric group is n!7824306f2b91
addedCayley's theorem (lead)743c0e710406
addedSymmetric group on a finite set10b962410330
addedSymmetric group of degree nf799f71a83b4
addedOrder is n!38c292a353c4
addedAbelian iff n ≤ 27ec1cd1855e6
addedTrivial for empty/singleton set4a7f8e5fba5b
addedSolvable iff n ≤ 4538feb852268
addedAbel–Ruffini consequence6633afe110ec
addedGalois group of general polynomial0184a9522cd6
addedCoxeter group of type A_n37f8107d267c
addedElements are permutations06c2e424cfc8
addedGroup operation is compositione7924ef3a1db
addedComposition of f and g0f2e1c34e406
addedCycle taken to k-th power13f8990b332b
addedClosure of composition1fe882c1860a
addedAssociativity of composition6b1da0140d12
addedIdentity element842517ec18c8
addedExistence of inverses83cf224db23a
addedTransposition280c6ecc4c24
addedEvery permutation is a product of transpositions1bbeaf8836c0
addedOdd and even permutations253046544d42
addedInvariance of parityc9adfb62b1be
addedParity of products3c5763e2c69d
addedSign of a permutation52ad0fbd320d
addedSign is a homomorphism655e5a51a5b9
addedAlternating group A_nf3617293d427
addedA_n normal of index 26306c32a9e72
addedS_n as semidirect productdc4a07d0dc37
addedProduct of adjacent transpositionsb36c5f99af97
addedCycle of length k8f4aa2149a0a
addedOrder of a cycle07b691198e29
addedLength-two cycles are transpositions267d8867493c
addedDisjoint cycles44ddd61f9938
addedDisjoint cycles commutee3a2fa2b5c66
addedDisjoint cycle decompositionbeeec71c712b
addedCycle conjugation propertydb85e0e39eeb
addedOrder reversing permutation1e7e63115825
addedLongest element / Bruhat order37a771562daa
addedReversal is an involutione1957fd137ab
addedPerfect shuffle2f53a0a981ea
addedConjugacy classes are cycle types9f63e605898a
addedConjugacy in S_5d8d4fa6fd10c
addedClasses correspond to partitions136dca4eb769
addedS_0 and S_1 are trivial6ab2984ab85c
addedStructure of S_2d890a9c82e45
addedS_3 is first nonabelian83219a6272b6
addedS_4 and rotations of the cube7860a881e025
addedKlein four-group in S_402124033c376
addedS_5 is first non-solvablef5a977ec83d4
addedS_6 has outer automorphismac6eac6489e6
addedTriple covers of A_6 and A_739ab4519b09c
addedNotable homomorphisms between symmetric groups238f8332d63a
addedA_n simple for n ≥ 50311767a9aa8
addedS_n semidirect product of A_n and S_2dd0dcdd9294c
addedAut(A_n) ≅ S_n for n ≠ 6c9bb1111e018
addedS_n is a complete group15972e913501
addedS_n almost simple for n ≥ 5c57ee863cb0c
addedEmbedding of S_n into A_{n+2}7583f18e365b
addedGenerated by adjacent transpositionsea8be3acbb38
addedCoxeter group structure83c1c6f76f9e
addedOther generating setsbe45e9532ac8
addedPermutation groupa81e064e6e78
addedNormal subgroups of finite S_n7e135d394b9b
addedNo index-2 subgroup on infinite setab26d99ab606
addedSchreier–Ulam theoremd7d1b58148ad
addedThree classes of maximal subgroups94059fa878a4
addedIntransitive maximal subgroups01a0b7824cc5
addedImprimitive maximal subgroupse4dff2ecf5b8
addedSylow subgroups are p-groups235e73ca7449
addedSylow p-subgroups of degree p4abab06f4ace
addedSylow p-subgroups of degree p^289edcf6df635
addedIterated wreath product W_p(n)7932b5c10dcf
addedSylow p-subgroups of degree n36d7a539fc20
addedTransitive subgroupcd96c34c2b0b
addedGalois group as transitive subgroupf6e42922fe4c
addedYoung subgroupaea26f895f24
addedYoung subgroups are parabolic940d1ff5d109
addedCayley's theorem7daca7869304
addedCyclic subgroup of a symmetric groupfd9178d51338
addedOrder is lcm of cycle lengths20ad96dd1af4
addedLargest element order is Landau's function6256245afd1b
addedS_n complete for n ≠ 2, 6f725b51ec9da
addedAutomorphism group for n = 2a79181878531
addedOuter automorphism for n = 6860481d99058
addedAutomorphisms inner except n = 69a2c061426a6
addedFirst homology of S_nae41152ea832
addedSecond homology (Schur multiplier)9507170de0e6
addedHomological stability49cceef2d522
addedHomology of infinite symmetric group1f3052026204
addedNumber of irreducibles equals partitionscd2b01e73917
addedIrreducibles realized over the integersde318ae700d3
addedMaschke semisimplicitye33d37ee59c3
addedSpecht modulesd8ea121c37e7