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Orders of finite perfect groups (groups such that G = G' where G' is the commutator subgroup of G).
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%I #17 Aug 23 2021 16:16:37

%S 1,60,120,168,336,360,504,660,720,960,1080,1092,1320,1344,1920,2160,

%T 2184,2448,2520,2688,3000,3420,3600,3840,4080,4860,4896,5040,5376,

%U 5616,5760,6048,6072,6840,7200,7500,7560,7680,7800,7920,9720,9828,10080,10752

%N Orders of finite perfect groups (groups such that G = G' where G' is the commutator subgroup of G).

%C This comment is about the three sequences A001034, A060793, A056866: The Feit-Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 [Corrected by _Isaac Saffold_, Aug 09 2021]

%C Since a non-cyclic simple group is perfect this sequence contains A001034 and since a perfect group is non-solvable this sequence is a subsequence of A056866 (apart from the initial term).

%D D. Holt and W. Plesken, Perfect Groups, Oxford University Press, 1989.

%H Eric M. Schmidt, <a href="/A060793/b060793.txt">Table of n, a(n) for n = 1..300</a>

%H Walter Feit, J. G. Thompson, <a href="http://www.pnas.org/content/48/6/968">A solvability criterion for finite groups and some consequences</a>, Proc. N. A. S. 48 (6) (1962) 968.

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%e A_{5} is perfect since it is equivalent to A_{5}'.

%o (GAP) SizesPerfectGroups(); # _Eric M. Schmidt_, Nov 14 2013

%Y Cf. A001034, A056866.

%K nonn

%O 1,2

%A Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 26 2001