A380006 Irregular triangle read by rows: T(n,k) is the number of non-isomorphic p-subgroups in the symmetric group S_n, where p is the k-th noncomposite divisor of n!.
1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 7, 2, 1, 1, 7, 2, 1, 1, 1, 34, 2, 1, 1, 1, 34, 7, 1, 1, 1, 61, 7, 2, 1, 1, 61, 7, 2, 1, 1, 1, 308, 12, 2, 1, 1
Offset: 1
Examples
Triangle T(n,k), with p = A008578(k), begins: n\p| 1 | 2 | 3 | 5 | 7 | ------------------------- 1 | 1 | | | | | 2 | 1 | 1 | | | | 3 | 1 | 1 | 1 | | | 4 | 1 | 4 | 1 | | | 5 | 1 | 4 | 1 | 1 | | 6 | 1 | 7 | 2 | 1 | | 7 | 1 | 7 | 2 | 1 | 1 | 8 | 1 | 34 | 2 | 1 | 1 |
Links
- A. Mann, Some questions about p-groups, Journal of the Australian Mathematical Society (Series A), 67(3) (1999), 356-379.
Programs
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GAP
LoadPackage("sonata"); A:=[]; for n in [1..9] do B:=[1]; G:=SymmetricGroup(n); for p in Primes do if p > Maximum(Factors(Factorial(n))) then break; fi; C:=[]; S:=ConjugacyClassesSubgroups(G); for m in [1..Length(S)] do if Unique(Factors(Order(Representative(S[m]))))=[p] and ForAll(C,K -> IsIsomorphicGroup(Representative(S[m]),K)=false) then C:=Concatenation(C,[Representative(S[m])]); fi; od; B:=Concatenation(B,[Length(C)]); od; A:=Concatenation(A,[B]); od; Print(Flat(A));
Extensions
Edited by Peter Munn, Mar 06 2025
Comments