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I am adding the definition of "Sporadic" to the article according to Gallian along with "The Monster" constructed by Griess and the Sylow Test for Nonsimplicity. I am unsure how to cite properly on the wiki, but here is the ISBN: 0-618-51471-6 page numbers are 420 to 423 for sporadic and 424 for the Sylow Test. Mrchapel0203 ( talk) 06:29, 5 September 2008 (UTC)
Even being familiar with the notion of a group, this article is hard to understand. A good explanation probably entails incorporating an example (rather than merely referring to it with a technical term) and actually showing why it is "simple". -- Beland 15:10, 18 December 2005 (UTC)
I removed the tag. I think enough effort has been expended on explaining some of the basic terms and motivation. Looking at the comments above, if you don't know what a subgroup is, there's only so much that is accessible in this article and there's really no reason it should be otherwise. By the way that guideline does not say "most articles should be geared...", it says technical articles should be made as accessible as possible, which is really the case here. Putting tags on articles like this only leads to dilution of effort as there are really problematic articles that are in far worse condition. -- C S (Talk) 04:49, 19 February 2007 (UTC)
I added a sentence about the Schreier conjecture. I seem to remember that the conjecture was originally stated for finite simple groups and has since been proved in that case, using the classification, and that the general case is outstanding, but I wanted to check it, and couldn't find it in my notes. -Lethe 02:32, 19 December 2005 (UTC)
Hi,
I am curious as if there is any special name for groups that, while not necessarily simple, cannot be expressed as the direct product of two or more smaller, non-trivial groups. All simple groups, I'm pretty sure, would satisfy this criterion, but a lot of non-simple groups would too. Examples include all cyclic groups of order p^n, where p is a prime number and n is an integer greater than 1 (if n=1 then the cyclic group would be of prime order and thus simple). Nonabelian examples include Dih3 and Dih4 (but not Dih6, which is isomorphic to the direct product of Dih3 and Z2). I'm primarily interested in finite groups but I imagine such a distinguishing property of groups exists for non-finite groups as well.
I'm surprised such a property of groups doesn't get more attention then it seems to. It seems a lot more obvious a distinguishing property of groups than simplicity as it is defined, and one could argue that finite groups that cannot be expressed as the direct product of multiple nontrivial groups, rather than the finite simple groups, are the basic building blocks of all finite groups. You can only create all finite groups from finite simple groups by allowing for semidirect rather than just direct products.
Thanks to whoever attempts to answer my question. Kevin Lamoreau 07:14, 20 January 2007 (UTC)
I'm not sure if this is the right place, or even if there really is any "right place" for this in wikipedia. However, a very common test for the nonsimplicity of a group of certain order is to find a subgroup of index n (typically by use of Sylow's theorem). If G is simple, then G is isomorphic to a subgroup of A_n. This argument can be used repeatedly since it is often the case that G is a subgroup of A_n for n >= 5, in such a case A_n is simple. So [A_n : G] = k, giving that A_n is isomorphic to a subgroup of A_k. For low order groups, it often turns out that n > k, which would be a contradiction.
Should this go into the article? —Preceding unsigned comment added by Rghthndsd ( talk • contribs) 07:12, 31 December 2008 (UTC)
I strongly vote for the inclusion of more tests for simplicity. Especially: if |G| = n*p^k and |G| does not divide n! then the group is not simple. Unfortunately my Wiki-fu is not up for the challenge of trying to write maths in a wiki article! 109.255.152.31 ( talk) 20:12, 28 April 2012 (UTC)
Like other abstract algebra pages (ex. nilpotent group#Definition), I think it would be great if this article had a list of "equivalent formulations." One equivalent formulation (that is hinted at in the conjugate closure article) is "the conjugate closure (also known as a normal closure) of every non-identity element is the whole group." Bender2k14 ( talk) 02:32, 28 February 2012 (UTC)
The introduction to this article states: "The complete classification of finite simple groups, completed in 2008, is a major milestone in the history of mathematics."
However, the "History for finite simple groups" section states re the classification program: " … and proof that this list was complete, which began in the 19th century, most significantly took place 1955 through 1983 (when victory was initially declared), but was only generally agreed to be finished in 2004."
So, were finite simple groups completely classified in 2004 or 2008 ?
Wikipedia's article "Classification of finite simple groups" says that the 2008 date is correct.
Cwkmail (
talk)
14:39, 31 August 2013 (UTC)
Isn't the definition for Lie groups different from that given here? I think that Lie groups are allowed to have nontrivial normal discrete subgroups. For instance, SL(2, C) has {I, −I} as a normal subgroup and it is simple. YohanN7 ( talk) 13:30, 21 April 2014 (UTC)
Does this mathematical concept have any real-world applications? Generally those are mentioned early in the article for the benefit of a wider audience. -- Beland ( talk) 08:35, 18 May 2024 (UTC)