A086792 Orders of finite groups G with the property that the sum of the orders of all the proper normal subgroups of G equals the order of G.

6, 12, 28, 30, 56, 360, 364, 380, 496, 760, 792, 900, 992, 1224, 1656, 1680, 1980

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 04 2003

Keywords

• nonn
• more

Comments

The only Abelian groups with this property are the cyclic groups C_n where n is a perfect number, so this sequence can be seen as a groups analogy of perfect numbers.

Derek Holt (mareg(AT)mimosa.csv.warwick.ac.uk) computed the orders of the non-Abelian groups in the sequence up to n=500 and commented "In general, if 2^n - 1 is a Mersenne prime, then 2^(n-1)*(2^n - 1) is a perfect number and the group with presentation < x,y | x^(2^n-1) = 1, y^(2^n) = 1, y^-1 x y = x^-1 > has order equal to the sum of the orders of its proper normal subgroups." So if n is an even perfect number, 2n also belongs to this sequence (the numbers 12 and 56 above).

Links

• Tom Leinster, Perfect numbers and groups, arXiv:math/0104012 [math.GR], 2001.
• Tom De Medts and Attila Maróti, Perfect numbers and finite groups
• Tom De Medts, Leinster groups. [archived]
• Sci.math, Groups analogy of perfect numbers

Crossrefs

Subsequence of A060652.

Cf. A000396.

Extensions

a(10)-a(17) added using "Leinster groups" link by Eric M. Schmidt, May 02 2014