A384800 -id:A384800 - cp's OEIS frontend

A368538 Integers k such that there exists a group of order k with exactly k subgroups.

1, 2, 6, 8, 28, 36, 40, 48, 54, 72, 96, 100, 104, 128, 132, 144, 160, 176, 180, 192, 216, 240, 252, 260, 288, 324, 336, 368, 384, 416, 456, 480, 496, 560, 576, 588, 624, 640, 672, 704, 720

Offset: 1

Robin Jones, Dec 29 2023

Powers of 4 cannot appear in this sequence. This is because for a group of order p^n, the number of subgroups of order p^k is congruent to 1 mod p, for 0 <= k <= n. It follows from p=2 and Lagrange's theorem that the number of subgroups of order 2^n for n even is congruent to 1 mod 2, i.e. not equal to 2^n. - Robin Jones, Feb 17 2024

a(34) >= 512. The smallest term strictly larger than 512 is 560. - Robin Jones, Feb 18 2024

			1 is a term since the trivial group (order 1) has exactly 1 subgroup.
2 is a term since the cyclic group C_2 has exactly 2 subgroups.
6 is a term since the symmetric group S_3 has exactly 6 subgroups.

Dave Benson, Congruence mod four of the number of subgroups of a finite 2-group, discussion in MathOverflow, Jun 11 2025.
Richard Stanley, What finite groups have as many elements as subgroups? Question in MathOverflow, answered by Dave Benson and others, Jun 07 2025.

Cf. A018216, A061034, A384727, A384800.

Missing term 36 added by Hugo Pfoertner, Jun 10 2025, following a suggestion by Dave Benson in the MathOverflow discussion.

a(34)-a(41) from Richard Stanley, Jun 11 2025, using results by Dave Benson in MathOverflow discussion of question 496010.

A384727 Number of groups of order n (up to isomorphism) with exactly n subgroups.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 1

Richard Stanley, Jun 08 2025

nonn

See A384800 for more information.

			The symmetric group S_3 has six elements and six subgroups. The other group of order six has four subgroups, so a(6)=1.

Richard Stanley, Table of n, a(n) for n = 1..192
Richard Stanley, What finite groups have as many elements as subgroups? Question in Mathoverflow, answered by Dave Benson and others, Jun 07 2025.

Cf. A368538, A384800.

cp's OEIS Frontend

A368538 Integers k such that there exists a group of order k with exactly k subgroups.

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