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A332766
Numbers k for which there exists a group of order k that cannot be generated by A051903(k) elements.
0
6, 10, 14, 18, 21, 22, 26, 30, 34, 36, 38, 39, 42, 46, 50, 54, 55, 57, 58, 62, 66, 70, 74, 78, 82, 86, 90, 93, 94, 98, 100, 102, 105, 106, 108, 110, 111, 114, 118, 122, 126, 129, 130, 134, 138, 142, 146, 147, 150, 154, 155, 158, 162, 165, 166, 170, 174, 178, 180
OFFSET
1,1
COMMENTS
For such k, it follows from the MathOverflow thread in Links that 1 + A051903(k) generators suffice.
Lists all k such that A059829(k) != A051903(k). Also k such that A059829(k) = 1 + A051903(k).
REFERENCES
R. Guralnick, A bound for the number of generators of a finite group, Arch. Math. 53 (1989), 521-523.
A. Lucchini, A bound on the number of generators of a finite group, Arch. Math. 53, (1989), 313-317.
EXAMPLE
k=20 is 2^2*5, so maximal exponent is 2. All five groups of order 20 can be generated by 2 elements. So 20 does NOT belong here.
On the other hand, k=21 is 3*7, so maximal exponent is 1. But there exists a group of order 21 that cannot be generated by 1 element. Therefore 21 belongs in this sequence.
CROSSREFS
Sequence in context: A284945 A091577 A115036 * A315169 A133653 A073760
KEYWORD
nonn
AUTHOR
Jeppe Stig Nielsen, Apr 25 2020
EXTENSIONS
More terms from Jinyuan Wang, Jun 26 2022
STATUS
approved