Wikipedia:Reference desk/Archives/Mathematics/2008 December 22

= December 22 =

What is the factorial of a fraction and a fractionth triangular number?
What is the factorial of a fraction and a fractionth triangular number? In other words I mean what is x and y when x=Πnundefinedi and y=∑nundefinedi where n is not an integer or n≠xINT1. Please help me! The Successor of Physics  08:25, 22 December 2008 (UTC) —Preceding unsigned comment added by Superwj5 (talk • contribs)
 * Neither of these functions are defined for fractional arguments, but there are fairly natural ways of extending them if for some reason you want to. The nth triangular number is n(n+1)/2, which gives you a definition that works on fractions, while the gamma function is the natural extension of factorial. Algebraist 08:31, 22 December 2008 (UTC)
 * I'd have said neither of these is defined. (Except that of course one can define then by using the Gamma function, although usually the language of factorials is not use then.) Michael Hardy (talk) 23:41, 22 December 2008 (UTC)
 * Yes, and also note that some people define a factorial function as $$x!:=\Gamma(x+1)$$, for real or complex x. PMajer --78.13.139.66 (talk) 08:51, 22 December 2008 (UTC)


 * What does n≠xINT1 mean? —Tamfang (talk) 17:27, 27 December 2008 (UTC)
 * I think it's a compsci way of saying n is not an integer. Algebraist 14:02, 29 December 2008 (UTC)
 * Weird if true. I might say . —Tamfang (talk) 00:38, 30 December 2008 (UTC)

Refs for group theory: realizability as derived or frattini subgroup
I remember there being a fair number of results about which groups can and cannot be the X-subgroup of a Y-group for various values of X and Y, but I don't remember really where all the results are proven or even just gathered. Values of X might include "derived", "frattini", "nilpotent residual", "solvable residual", and values of Y might include "any", "finitely generated", "finite", "solvable", "nilpotent".

As an example it is easy to show that the nonabelian group of order six is not the derived subgroup of any group (complete group), nor can a nonabelian group of order eight be the Frattini subgroup of any finite group (contradiction when looking in the automorphism group).

Can someone find a reasonable survey of these types of results? I am happy if X and Y are fixed in the survey, or if the survey covers multiple values of X and Y.

I am particularly interested in complete classifications for X=derived, Y=finite, and X=frattini, Y=finite. JackSchmidt (talk) 15:47, 22 December 2008 (UTC)
 * One place to start would be B. Eick's "The converse of a theorem of W. Gaschutz on Frattini subgroups" in Math Z. #224. I don't have access to a copy right now, but it and the references it cites might be helpful. 86.15.139.246 (talk) 23:13, 22 December 2008 (UTC)
 * Excellent, thanks! This is probably the main source I was remembering.  I read this one a few years ago, but not recently.  It shows that a finite group is isomorphic to a subgroup of the Frattini group of a Y-group if and only if it is a nilpotent Y-group (where Y is "any" or "finite" or "p" etc.).  The condition of Gaschütz itself is a little hard to use abstractly, but for a specific example is probably reasonable enough, and it (now) characterizes the finite groups that are isomorphic to the Frattini subgroup of a Y-group (where Y is "any" or "finite" or "solvable" or "nilpotent").  Section 4 works out a similar idea for the derived subgroup, but the result is only one way (if the group is isomorphic to a derived subgroup, then it satisfies this condition).  JackSchmidt (talk) 01:15, 23 December 2008 (UTC)

Probability
I have set set of words, T, containing x number of words. The teacher picks a subset of T, containing y number of words (I don't know which words). I must know the definitions of z number of words of the y number of words the teacher picked. How would I set this up to know the least number of words whose definitions I must memorize to be sure to know the z number of words from y? TIA, Ζρς ι'β' ¡hábleme! 21:56, 22 December 2008 (UTC)
 * The number of words you needn't know is $$y-z$$. So the neumber of words you do need to know is $$x-(y-z) = x+z-y$$ —Preceding unsigned comment added by Philc 0780 (talk • contribs) 22:13, 22 December 2008