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

1,8

Comments

In the theory of finite p-groups it can be asked: how many p-groups can be faithfully represented as permutation groups of degree n? This is equivalent to asking how many p-subgroups does the symmetric group S_n contain, up to isomorphism.

T(n,1) is included as the trivial group is considered a p-group.

Links

Table of n, a(n) for n=1..49.

A. Mann, Some questions about p-groups, Journal of the Australian Mathematical Society (Series A), 67(3) (1999), 356-379.

Example

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 |

Prog

(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));

Crossrefs

Cf. A000142, A008578, A036234 (row lengths).

Sequence in context: A316584 A362856 A381706 * A146325 A333807 A069289

Adjacent sequences: A380003 A380004 A380005 * A380007 A380008 A380009

Keyword

nonn,tabf,more,changed

Author

Miles Englezou, Jan 07 2025

Extensions

Edited by Peter Munn, Mar 06 2025

Status

approved