Talk:Simple group

From Wikipedia, the free encyclopedia

[edit] 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)

Wikipedia: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. --Chan-Ho (Talk) 04:49, 19 February 2007 (UTC)

[edit] 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)

[edit] 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)