A332766 Numbers k for which there exists a group of order k that cannot be generated by A051903(k) elements.

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.

Links

Table of n, a(n) for n=1..59.

MathOverflow, How large can the smallest generating set of a group G of order n be?

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

Cf. A051903, A059829.

Sequence in context: A284945 A091577 A115036 * A315169 A133653 A073760

Adjacent sequences: A332763 A332764 A332765 * A332767 A332768 A332769

Keyword

nonn

Author

Jeppe Stig Nielsen, Apr 25 2020

Extensions

More terms from Jinyuan Wang, Jun 26 2022

Status

approved