Talk:Simple group

Sporadic Simple Groups
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)


 * Thanks. I cleaned up the wiki-conventions and added a little detail.  I'll check if Gallian supports what is now claimed, or just use a standard classification reference. JackSchmidt (talk) 14:25, 5 September 2008 (UTC)


 * Thanks for the help with the wiki-conventions. With just a quick compare between the current version, my revision and Gallian, it looks like everything is up to par. —Preceding unsigned comment added by Mrchapel0203 (talk • contribs) 21:17, 14 September 2008 (UTC)

Request for technical explanation
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 added some examples. Let me know what you think. Cheers, Oleg Alexandrov (talk) 19:31, 18 December 2005 (UTC)


 * Thank you for the additional material...it's definitely getting better. Just reading the article once through, it's unclear to me what "G=Z/3Z" is.  I eventually figured it out by reading subarticles, but I think actually explicitly writing out the constituent parts of G and perhaps candidates for H would make it immediately clear, even if you have only a general idea of what a group is. Noting "commutative" as an alternate term for "abelian" would probably save a lot of people from having to look up that term.  It would also definitely be helpful to have a quick gloss on the definition of normal.  A group by definition forms a closed system; is a normal subgroup just one which is also a closed system, or is there something slightly more to it than that?  The notation in normal subgroup is also somewhat opaque. -- Beland 05:21, 22 December 2005 (UTC)
 * Now I will argue that you are asking a bit too much. This article is not a textbook on group theory starting from zero. One should not attempt to read this article if one does not know what Z/3Z and a normal subroup are, and that commutative is the same as abelian. The name "simple group" is deceptive, these are very complicated beasts. Oleg Alexandrov (talk) 06:04, 22 December 2005 (UTC)
 * Make technical articles accessible says that most articles should be geared toward a general audicence, a far cry from "people who already know what G=Z/3Z is". That might not be feasible, but a little extra time spent on explaining basic terms would make it more accessible to readers who understand math on the level of a typical MIT graduate, like me. -- Beland 05:41, 26 December 2006 (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)

schreier conjecture
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)

Groups that cannot be expressed as the direct product of other groups
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)


 * They are called indecomposable groups (or directly indecomposable groups if there is any danger of confusion with other types of indecomposability). We don't seem to have an article on indecomposable groups, but take a look at Krull-Schmidt theorem. Part of the reason that they don't receive as much attention as simple groups is that you can't say very much about them - they are much more varied than simple groups. (By the way, the situation for simple groups is worse than you suggest: semidirect products do not suffice to build all finite groups from finite simple groups.) --Zundark 10:02, 21 January 2007 (UTC)


 * Wow! That information is great. Thanks! Kevin Lamoreau 16:28, 21 January 2007 (UTC)

Tests for nonsimplicity
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)

Equivalent formulations of simple groups
Like other abstract algebra pages (ex. nilpotent group), 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)

Date of completion of classification ?
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)

Infinite simple groups
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)

Applications
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)