|
%I #20 Nov 10 2022 16:11:30
%S 6,12,28,30,56,360,364,380,496,760,792,900,992,1224,1656,1680,1980
%N 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.
%C 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.
%C 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).
%H Tom Leinster, <a href="https://arxiv.org/abs/math/0104012">Perfect numbers and groups</a>, arXiv:math/0104012 [math.GR], 2001.
%H Tom De Medts and Attila Maróti, <a href="http://cage.ugent.be/~tdemedts/preprints/leinster.pdf">Perfect numbers and finite groups</a>
%H Tom De Medts, <a href="http://java.ugent.be/~tdemedts/leinster/">Leinster groups</a>. [<a href="https://web.archive.org/web/20180820105225/http://java.ugent.be/~tdemedts/leinster/">archived</a>]
%H Sci.math, <a href="http://mathforum.org/kb/thread.jspa?forumID=13&threadID=97444">Groups analogy of perfect numbers</a>
%Y Subsequence of A060652.
%Y Cf. A000396.
%K nonn,more
%O 1,1
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 04 2003
%E a(10)-a(17) added using "Leinster groups" link by _Eric M. Schmidt_, May 02 2014
|
