Following on from some of myprevious MO questions on finite group theory...
$\newcommand{\Irr}{\operatorname{Irr}}\newcommand{\Conj}{\operatorname{Conj}}\newcommand{\AMZL}{{\rm AM}\_{\rm Z}}$


Let $G$ be a finite group, $\Irr$ the set of irreducible characters (working over $\mathbb C$) and $\Conj$ the set of conjugacy classes. Consider the following quantity, which seems to have first been introduced in work of [Azimifard, Samei and Spronk](http://arxiv.org/abs/0805.3685)
$$\alpha\_G = \frac{1}{|G|^2} \sum\_{\phi\in\Irr} \sum\_{C\in\Conj} \phi(e)^2 |\phi(C)|^2 |C|^2 . $$
It is not hard to show, just using basic character theory, that $\alpha\_G \geq 1$ with equality if and only if $G$ is abelian. This constant/invariant arises as a minorant for a more interesting invariant of $G$ which was studied in the aforementioned paper.


Call this "more interesting invariant" $\AMZL(G)$. It turns out that $\AMZL(G)\geq \AMZL(G/N)$ for every normal subgroup $N$, a property which is vital to some work I am writing up that obtains lower bounds on $\AMZL(G)$. One branch of the case-by-case attack I'm using works by getting lower bounds on $\alpha\_G$ when $G$ has trivial centre, and so if $\alpha\_G$ never increases when we replace $G$ with a quotient then I could in fact get general lower bounds on $\alpha\_G$ by inductively modding out by the centre.



> 
> **Question.** Do there exists a finite group G and a normal subgroup N for which $\alpha\_{(G/N)} > \alpha\_G$ ?
> 
> 
> 


I did try for a while to show this can never happen but rapidly got stuck, so I'm hoping that MO readers who are more skilled/experienced with finite groups can either suggest counterexamples to try, or give some evidence that no such counterexamples exist. Please bear in mind that I have next to no experience with GAP or similar, so I don't mind concrete suggestions of code, but responses of the form "get GAP to look through all non-simple non-abelian examples of order < 100" may not be as helpful as their authors imagine.




---


Just in case it helps to rule out certain avenues of attack: if $G$ is a group with two character degrees (i.e. there is an integer $m$ such that $\phi(e)\in \{1,m\}$ for all $\phi\in\Irr$) then one can evaluate $\alpha\_G$ explicitly, using ideas similar to [this paper](http://arxiv.org/abs/1302.1929): one obtains


$$
\alpha\_G - 1 = (m^2-1) \left( 1- \frac{|L|}{|G|^2} \sum\_{C\in\Conj} |C|^2 \right)
$$
where $|L|$ is the number of linear characters on $G$.


In particular, suppose if $G=H \rtimes C\_2$ is a generalized dihedral group with $|H|=2n+1$ ($n$ a positive integer). Back of the envelope scribbling gives me $m=2$, $|L|=2$ and the conjugacy classes have sizes $1$, $2$ repeated $n$ times, and $2n+1$, so that
$$ \frac{\alpha\_G -1}{3} = 1 - \frac{1}{2(2n+1)^2}(1+ 4n + (2n+1)^2) = \frac{1}{2}\left(1-\frac{1}{2n+1}\right)^2 $$
It also seems to me that proper quotients of $G$ must either be abelian or generalized dihedral with smaller order, in which case we would have $\alpha\_G \geq \alpha\_{(G/N)}$ for every $N \lhd G$.



A constant associated to the character table of a finite group
Let $G= \operatorname{SL}(2,5)$ and $N= \mathbf{Z}(G)$, so that $G/N = \operatorname{PSL}(2,5)\cong A\_5$. If my code works correctly, then 
$$ \alpha\_{G/N} = 842/75 > 6661/600 = \alpha\_G. $$ 


My implementation of alpha (for character tables) in GAP: 



```
alpha:= function( tbl )
    local clssizes, n, kg, phi, i;
    clssizes:= SizesConjugacyClasses( tbl );
    n:= Size( tbl );      # group order
    kg:= NrConjugacyClasses( tbl );
    return Sum( Irr( tbl ), 
                phi-> phi[1]^2 
                      * Sum( [1..kg], 
                             i-> phi[i] * ComplexConjugate( phi[i] )
                                        * clssizes[i]^2 
                           )
              )/ n^2 ;
end;

```

