Comments on: Groups of Order Sixteen http://www.arsmathematica.net/archives/2008/04/08/652/ Dedicated to the mathematical arts. Tue, 02 Dec 2008 01:54:45 +0000 http://wordpress.org/?v=2.5.1 By: Jacob Freeze http://www.arsmathematica.net/archives/2008/04/08/652/#comment-59665 Jacob Freeze Wed, 16 Apr 2008 22:15:12 +0000 http://www.arsmathematica.net/archives/2008/04/08/652/#comment-59665 A few links into the references on this post turned up an unfamiliar way of appreciating the rarity of non-solvable groups: If you call a number "solvable" if every group of that order is solvable, then... <a href="http://www.research.att.com/~njas/sequences/A056866" rel="nofollow">A positive integer n is a non-solvable number if and only if it is a multiple of any of the following numbers: a) 2^p(2^2p-1), p any prime. b) 3^p(3^2p-1)/2, p odd prime. c) p(p^2-1)/2, p prime greater than 3 such that p^2+1 = 0 (mod 5). d) 2^4*3^3*13. e) 2^2p(2^2p+1)(2^p-1), p odd prime. </a> A few links into the references on this post turned up an unfamiliar way of appreciating the rarity of non-solvable groups:

If you call a number “solvable” if every group of that order is solvable, then…

A positive integer n is a non-solvable number if and only if it is a multiple of any of the following numbers: a) 2^p(2^2p-1), p any prime. b) 3^p(3^2p-1)/2, p odd prime. c) p(p^2-1)/2, p prime greater than 3 such that p^2+1 = 0 (mod 5). d) 2^4*3^3*13. e) 2^2p(2^2p+1)(2^p-1), p odd prime.

]]> By: Jan http://www.arsmathematica.net/archives/2008/04/08/652/#comment-59657 Jan Tue, 15 Apr 2008 21:55:33 +0000 http://www.arsmathematica.net/archives/2008/04/08/652/#comment-59657 Just curious, but who is the author of this great blog? You should make yourself known. Just curious, but who is the author of this great blog? You should make yourself known.

]]> By: jimcp http://www.arsmathematica.net/archives/2008/04/08/652/#comment-59650 jimcp Sat, 12 Apr 2008 10:36:59 +0000 http://www.arsmathematica.net/archives/2008/04/08/652/#comment-59650 Another thank you for pointing to this article. Another thank you for pointing to this article.

]]> By: John Cook http://www.arsmathematica.net/archives/2008/04/08/652/#comment-59631 John Cook Wed, 09 Apr 2008 10:18:28 +0000 http://www.arsmathematica.net/archives/2008/04/08/652/#comment-59631 Thanks for pointing out the article. Simplifying previous results is not often rewarded professionally, but is greatly appreciated. Thanks for pointing out the article. Simplifying previous results is not often rewarded professionally, but is greatly appreciated.

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