

A001034


Orders of noncyclic simple groups (without repetition).
(Formerly M5318 N2311)


22



60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 12180, 14880, 20160, 25308, 25920, 29120, 32736, 34440, 39732, 51888, 58800, 62400, 74412, 95040, 102660, 113460, 126000, 150348, 175560, 178920
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OFFSET

1,1


COMMENTS

This comment is about the three sequences A001034, A060793, A056866: The FeitThompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even.  Ahmed Fares (ahmedfares(AT)mydeja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]
The primitive elements are A257146. These are also the primitive elements of A056866.  Charles R Greathouse IV, Jan 19 2017
Conjecture: This is a subsequence of A083207 (Zumkeller numbers). Verified for n <= 156. A fast provisional test was used, based on Proposition 17 from Rao/Peng JNT paper (see A083207). For those few cases where the fast test failed (such as 2588772 and 11332452) the comprehensive (but much slower) test by T. D. Noe at A083207 was used for result confirmation.  Ivan N. Ianakiev, Jan 11 2020
From M. Farrokhi D. G., Aug 11 2020: (Start)
The conjecture is not true. The smallest and the only counterexample among the first 457 terms of the sequence is a(175) = 138297600.
On the other hand, the orders of sporadic simple groups are Zumkeller. And with the exception of the smallest two orders 7920 and 95040, the odd part of the other orders are also Zumkeller. (End)
Any group G whose order is 2*n, where n is odd, has an element of order two, by Cauchy's Theorem. This element's image under G's left (or right) regular representation is not in the alternating group, and so G has a normal subgroup of index two. When considered along with the FeitThompson theorem, this shows that all terms of this sequence are divisible by four.  Isaac Saffold, Aug 09 2021


REFERENCES

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
Dickson L.E. Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.
M. Hall, Jr., A search for simple groups of order less than one million, pp. 137168 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

M. Farrokhi D. G., Table of n, a(n) for n = 1..10000
C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574577.
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory (page images), Dover, NY, 1958, p. 309.
M. Farrokhi D. G., A GAP function generating the smallest noncyclic finite simple group order greater than a given number m.
Walter Feit and J. G. Thompson, A solvability criterion for finite groups and some consequences, Proc. N. A. S. 48 (6) (1962) 968.
M. Hall Jr., Simple groups of order less than one million, J. Alg. 20 (1) (1972) 98102
David A. Madore, More terms
Index entries for sequences related to groups
Index entries for "core" sequences


CROSSREFS

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.
Cf. A109379 (orders with repetition), A119648 (orders that are repeated).
Sequence in context: A329521 A118671 A109379 * A330583 A330585 A330584
Adjacent sequences: A001031 A001032 A001033 * A001035 A001036 A001037


KEYWORD

nonn,nice,core


AUTHOR

N. J. A. Sloane, Simon Plouffe


STATUS

approved



