Talk:Central series

Confusing

I am finding this article very hard to understand. Here are two of my problems:

1. The section Upper central series uses something called "E", which is nowhere defined. Maybe it means the 1-element group. But in the Definition section, that is called "1". If E means 1, why not just use the same name throughout the article? If it means something different, readers should be told what.
2. The first sentence of the Definition section refers to "a normal series", and the second sentence to "a subnormal series". According to Wikipedia these terms mean the same (so it is confusing to switch names from one sentence to the next). But Humphreys in "A Course in Group Theory" distinguishes between the two. It is not clear what is meant by them in this article.

Maproom (talk) 13:47, 16 September 2008 (UTC)

1 is fixed.

2 is a little more complicated. I'll fix it in a bit.

So by "normal series", this article means the strong definition, also known as "invariant series". That is, a finite sequence of normal subgroups that form a chain under inclusion. By subnormal series it means a finite sequence of subnormal subgroups that form a chain under inclusion. In order to talk about G/Ai at all, Ai should be normal in G. However, to talk about [G,A(i+1)], requires nothing about Ai. Instead of requiring it to be subnormal, I'll require nothing. I'll also specify which definition of normal series is in use. JackSchmidt (talk) 14:58, 16 September 2008 (UTC)

Ok 2 fixed. BTW if you are just trying to understand this stuff, then start with just one of the central series. The LCS behaves well in quotient groups, but only requires subgroups to define. The UCS behaves well in subgroups, but requires quotient groups to define. I recommend the LCS if you have no preference. JackSchmidt (talk) 15:13, 16 September 2008 (UTC)

LCS and UCS are both defined using commutators, without any explanation of the relationship between centres and commutators - this does not make things easy.

Under Lower Central Series, the article says "This should not be confused with the derived series, whose terms are G⁽ⁿ⁾ := [G⁽ⁿ⁻¹⁾,G⁽ⁿ⁻¹⁾], not G_n := [G_n−1,G]."

1. I assume that these superscripts denote members of the series, just as subscripts would. Am I right? Is there a reason for the switch to superscripts?
2. Two of these superscripts include n, one includes n. Does this signify something? Maproom (talk) 17:53, 26 September 2008 (UTC)

I'll assume you are mostly asking about learning math, rather than asking about the wikipedia policy. The wikipedia answer is very dull: this is how the sources do it / wikipedia is filled with typographical errors. Here is the learning math answer:

For center vs. commutator, I'm not sure what explanation there should be? An element x of G is in the center of G if and only if xy=yx for all y in G, if and only if [x,y]=1 for all y in G (both definitions are given in the article). Being in the center means commuting and commutators measure commuting.

(1) Yes, they denote a sequence of subgroups. Both upper and lower indexed sequences are extremely common in algebra. Often, but certainly not always, superscripts are enclosed in parentheses to avoid confusion with various powering operations. The derived series notation mimics the notation for derivatives. The derived subgroup is G′, just like the derivative is f′. The second (proper) term of the derived series is G′′ just like the second derivative is f′′. Certainly by the fourth or fifth derivative, instead of repeating primes, one uses G⁽⁵⁾ for the fifth (proper) term of the derived series and f⁽⁵⁾ for the fifth derivative. For various formulas like the Taylor series, it is handy to have a 0th derivative, f = f⁽⁰⁾ and G=G⁽⁰⁾. Note that the lower central series descends as its index increases from 1, the derived series descends as its index increases from 0, and the upper central series ascends as its index increases from 0. Keeping track of all of those conventions is a pain, but they are all chosen well. (2) No significance, all of the n should be italicized. I fixed the ones I saw. JackSchmidt (talk) 18:23, 26 September 2008 (UTC)

I am trying to learn the math from these pages - and I feel that if I can't make sense of them, maybe others can't either, and it's at least partly the page's fault. I am getting there now, thanks to your improvements and explanations; and I have grasped some of the connection between commutators and centres.

If I see a typo in a page about say the Battle of Blenheim, I correct it, even though I am no historian. But when it's in a page of math, I can't be sure it's a typo, hence my more trivial questions.

I can see a problem which must be common on Wikipedia. When someone writes a book, or a lecture course, on group theory, he uses terms like "subnormal" and "ring" consistently; so when you jump from chapter to chapter or lecture to lecture, you know what the words mean. But on Wikipedia, they can mean different things in different articles, without any warning. I guess this is inevitable. Maproom (talk) 21:46, 29 September 2008 (UTC)

Ok, I'm slowly understanding this. I would find some examples helpful. If I have got things right:

• A central series for S4 does not exist.
• The lower central series for S4 goes S4, A4, A4, A4, ...
• The upper central series for S4 goes 1, 1, 1, ... Maproom (talk) 12:03, 4 December 2008 (UTC)