cp's OEIS Frontend

All such orders are composite numbers (since there is only one group of any prime order).

Orders which are repeated in A109379.

Except for the first number, these are the orders of symplectic groups C_n(q)=Sp_{2n}(q), where n>2 and q is a power of an odd prime number (q=3,5,7,9,11,...). Also these are the orders of orthogonal groups B_n(q). - Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010

a(1) = 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8 (see A137863). - Bernard Schott, May 18 2020

From Bernard Schott, Apr 26 2020: (Start)

About a(16) = 20160; 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but, 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8.

Indeed, 20160 is the smallest order for which there exist two nonisomorphic simple groups and it is the order of this group PSL_3(4) that was missing in the data. The first proof that there exist two nonisomorphic simple groups of this order was given by the American mathematician Ida May Schottenfels (1900) [see the link]. (End)

The other names for groups of this kind include "semidirectly indecomposable groups" or "inseparable groups". Note that the following are equivalent definitions for a nontrivial group to be a splitting-simple group:

- It is not the (internal) semidirect product of proper subgroups;

- It is not isomorphic to the (external) semidirect product of nontrivial groups;

- It has no proper nontrivial normal subgroups with a permutable complement.

- It is the non-split extension of every proper nontrivial normal subgroup by the corresponding quotient group.

Also note that being simple is a stronger condition than being splitting-simple, while being directly indecomposable (see A090751) is weaker.

a(p^e) >= 1 since C_p^e cannot be written as the semidirect product of proper subgroups. For e >= 3, a(2^e) >= 2 by the existence of the generalized quaternion group of order 2^e, which is the only non-split extension of C_2^(e-1) by C_2 other than C_2^e.

The smallest numbers here with a(n) > 0 that are not prime powers are 48, 60, 120, 144, 168, 192, 240, 320, 336, 360 and so on. Are there any odd numbers n that are not prime powers satisfying a(n) > 0 ?

Conjecture: a(n) = 0 for squarefree n which is not a prime.

The conjecture that a(n) = 0 for nonprime squarefree n is true. Proof: It is known that every group G of squarefree order is supersolvable; hence G contains a normal series with prime cyclic factors. Since every Sylow subgroup of G is prime cyclic, these cyclic factors are isomorphic to the Sylow subgroups of G. Let P be one such factor; then for an appropriate M in G, P = G/M, where |G| = |P|*|M|. By the Schur-Zassenhaus theorem, G is a semidirect product of M and P, and a(n) = 0 when n is squarefree. - Miles Englezou, Oct 24 2024

Equivalently, numbers k > 1 such that there exists a group of order k with Sylow number > 1 for every prime dividing k.

The corresponding numbers of groups of order k without nontrivial normal Sylow subgroups are: 1, 4, 1, 4, 17, 3, 17, 1, 1, 1, 86, 18, 8, 90, 1, 5, 1, 3, 6, 536, 80, 27, ...

Note that if G has no normal Sylow p-subgroups, p divides |G|, then G X C_p also has no normal Sylow p-subgroups. That is to say, if k is in this sequence and p divides k, then k*p is also in this sequence. In particular, every number of the form 24 * 2^a * 3^b * 5^c * 7^d with nonnegative a,b,c,d is here.

The "primitive" terms (the terms not of the form k*p where k is a previous term and p divides k) are 24, 60, 160, 168, 324, 660, 784, 840, 896, 1053, ...

Includes A001034 as a subsequence, by the definition of non-cyclic simple groups. If q is not a prime power (not in A246655) and A338757(q) > 0, then q is here, as guaranteed by Schur-Zassenhaus theorem.

If k = p^e * q is here, p, q distinct primes, then q == 1 (mod p) and e >= 1+ord(p,q), where ord(p,q) is the multiplicative order of p modulo q. Proof: Let G be a group of order k without nontrivial normal Sylow subgroups. Let n_p (respectively n_q) be the number of Sylow p-subgroups (respectively q-subgroups), then n_p, n_q > 1. By Sylow's 3rd theorem, we have n_p == 1 (mod p), n_p | q; n_q == 1 (mod q), n_q | p^e. It is possible only if q == 1 (mod p) and e >= ord(p,q).

If e = ord(p,q), then we must have n_q = p^e. The Sylow q-subgroups have order q, which is a prime, so the pairwise intersections must be trivial, i.e., there are p^e * (q-1) = k - p^e elements in G of order q. The remaining p^e elements are just enough to make a unique Sylow p-subgroup, so n_p = 1, which is a contradiction. Hence, e >= 1+ord(p,q).

The terms of the form p^e * q where e = 1+ord(p,q), q == 1 (mod p) are 24 = 2^3 * 3, 160 = 2^5 * 5, 1053 = 3^4 * 13 and so on. Note that q == 1 (mod p) and e >= 1+ord(p,q) are only necessary but not sufficient: 112 = 2^4 * 7 satisfies 7 == 1 (mod 2) and 4 >= 1+ord(2,7), but 112 is not here. Similarly, 19375 = 5^4 * 31 satisfies 31 == 1 (mod 5) and 4 >= 1+ord(5,31), but 19375 is not here.

By Lagrange's Theorem a(n) is a multiple of n.

This includes all the sporadic simple groups (A001228) except one (the Monster).

All orders of nonsolvable groups A056866 are divisible by 4.

When considering whether an integer k is the order of a finite simple group, the first thing one checks is whether the number of p-Sylow subgroups is forced to be 1 for some p dividing k. This occurs if the only divisor of k which is 1 (mod p) is 1 itself. This sequence consists of the numbers that survive this test.

See A348305 for the basic definitions and references about fusion rings, or page 60 in Etingof et al. The "rank" of a fusion ring is the cardinal of its basis. The Frobenius-Perron dimension (FPdim) of a fusion ring is the sum of the square of the FPdim of its basic elements.

A fusion ring is called "integral" if the FPdim of its basic elements are integers. The group ring ZG is an example of integral fusion ring, where the finite group G is the basis. The character ring ch(G) is another example of integral fusion ring, where the basis is the set of irreducible characters.

A fusion ring is called "simple" if it has no proper nontrivial fusion subring. The fusion ring ch(G) is simple iff the finite group G is simple.

The fusion ring ch(G) remembers the simple group G (not true for non-simple groups, e.g., D4 and Q8), moreover there are plenty of simple integral fusion rings not of the form ch(G). So a classification of simple integral fusion rings would "really" extend CFSG.

The minimal rank for a non-pointed simple integral fusion ring of the form ch(G) is 5, given by G=A5. But in general, the minimal rank for a simple integral fusion ring is 4, as proved in Alekseyev et al.

The list of twelve simple integral fusion rings of rank 4 and FPdim < 10^6 is available in slide 19 of the talk "Exotic Integral Quantum Symmetry" in the Links section.

A group G is almost simple if there exists a (non-abelian) simple group S for which S <= G <= Aut(S).