A274847 a(n) = number of similarity classes of groups with exactly n subgroups (see reference for precise definition of similarity classes).

1, 1, 1, 2, 2, 5, 1, 7, 2, 12, 4, 11, 1, 17, 8, 22, 3, 22, 5

Offset: 1

Author

Michael C Slattery, Jul 08 2016

Keywords

• nonn
• more

Comments

See Slatterly references for precise definition of similarity classes and a proof of the first 12 terms.

See Betz and Nash for correction of a(10) and proof of terms 13-19.

Examples

			For n = 6 the a(6) = 5 similarity classes of groups with 6 subgroups are Z_{p^5}, Z_p X Z_{q^2}, Z_3 X Z_3, S_3, Q_8.
		

Links

• Alexander Betz and David A. Nash, Classifying groups with a small number of subgroups, arXiv:2006.11315 [math.GR], 2020.
• Angsuman Das and Arnab Mandal, Solvability of a group based on its number of subgroups, arXiv:2403.01262 [math.GR], 2024.
• George A. Miller, Groups having a small number of subgroups, Proc. Natl. Acad. Sci. U S A, vol. 25 (1939) 367-371.
• David A. Nash and Alexander Betz, Classifying groups with a small number of subgroups, arXiv:2006.11315 [math.GR], 2020.
• Michael C. Slattery, On a property motivated by groups with a specified number of subgroups, Amer. Math. Monthly, 123 (2016), 78-81.
• Michael C. Slattery, Groups with at most twelve subgroups, arXiv:1607.01834 [math.GR], 2016.

Crossrefs

Cf. A018216, A289445.

Extensions

Correction of a(10) and extension to 19 terms by David A. Nash, Jun 29 2020