Let $G$ be a finite group, $L(G)$ its subgroup lattice and $\mu$ the Möbius function.  

Consider the Euler totient of $G$ defined as follows:  

$$ \varphi(G) = \sum\_{H \le G}\mu(H,G) |H| $$
Let $X=\{M\_1, \dots, M\_n \}$ be the set of maximal subgroups of $G$. By applying the *Crosscut Theorem* with $X$ (see [this comment](https://mathoverflow.net/questions/264107/a-boolean-representation-of-the-m%C3%B6bius-function-on-a-finite-lattice#comment651597_264107) of Richard Stanley) and next the [inclusion–exclusion principle](https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle), we get that:
$$ \varphi(G) = |G \setminus \bigcup\_{i=1}^n M\_i| $$ In other words, $\varphi(G)$ is the number of elements $g \in G$ such that $\langle g \rangle = G$. It follows that 
$$ \varphi(G) \neq 0 \Leftrightarrow G \text{ cyclic}$$


Note that $\varphi(\mathbb{Z}/n) = \varphi(n)$ the usual [Euler's totient function](https://en.wikipedia.org/wiki/Euler's_totient_function).


Now, consider the dual Euler totient of $G$ defined as follows: 
$$ \hat{\varphi}(G) = \sum\_{H \le G}\mu(1,H) |G:H| $$ 


**Question**: $ \hat{\varphi}(G) \neq 0 \Leftrightarrow G$ has a faithful irreducible complex representation?


*Remark*: We will see below that $(\Rightarrow)$ is true. So the question reduces to $(\Leftarrow)$.  

It is true for the finite simple group $G$ of order $<10000$: 


$$ \begin{array}{c|c|c|c|c|c}
G & |G| & \hat{\varphi}(G) \newline 
 \hline 
 A\_5 & 60 & 8 & \newline 
 \hline
 PSL(2,7) & 168 & 228 & \newline 
 \hline
 A\_6 & 360 & 8748 & \newline 
 \hline
 PSL(2,8) & 504 & 19056 & \newline 
 \hline
PSL(2,11) & 660 & 24932 & \newline 
 \hline
PSL(2,13) & 1092 & 105684 & \newline 
 \hline
 PSL(2,17) & 2448 & 389496 & \newline 
 \hline
 A\_7 & 2520 & 188136 & \newline 
 \hline
 PSL(2,19)& 3420 & 1148028 & \newline 
 \hline
 PSL(2,16)& 4080 & 1935584 & \newline 
 \hline
 PSL(3,3)& 5616 & 395496 & \newline
 \hline
 PSU(3,3)& 6048 & 507168 & \newline
 \hline
 PSL(2,23)& 6072 & 2234784 & \newline
 \hline
 PSL(2,25)& 7800 & 5391800 & \newline
 \hline
 M\_{11} & 7920 & 1044192 & \newline
 \hline
 PSL(2,27)& 9828 & 7778916 & \newline
\end{array}$$ 


Any idea about the meaning of these numbers?




---


*Proof of $(\Rightarrow)$*


Consider the relative version $$ \hat{\varphi}(H,G) = \sum\_{K \in [H,G]}\mu(H,K) |G:K|.$$ *Warning*: $-\hat{\varphi}(H,G)$ is **not** the Möbius invariant of the bounded coset poset $\hat{C}(H,G)$ because $$-\mu(\hat{C}(H,G)) = \sum\_{K \in [H,G]}\mu(K,G) |G:K| $$ and $\mu(K,G) \neq \mu(H,K)$ in general. 


Now if $[H,G]$ is boolean of rank $n$ then $\mu(K,G) = (-1)^n \mu(H,K)$; moreover (independently) by Theorem 3.21 of [this paper](https://arxiv.org/pdf/1604.06765.pdf), if $ \hat{\varphi}(H,G) \neq 0$ then there is an irreducible complex representation $V$ of $G$ such that $G\_{(V^H)} = H$. Next, using a dual reformulation of the Crosscut Theorem with $X$ the set of atoms, we can extend the proof of Theorem 3.21 to any interval $[H,G]$ (i.e. without assuming it to be boolean). Finally by taking $H = 1$, we get that for any finite group $G$, if $\hat{\varphi}(G) \neq 0$ then there is an irreducible complex representation $V$ such that $G\_{(V)} = 1$, which means that $V$ is faithful. 



Existence of a faithful irreducible representation using Möbius function
No, the modular maximal-cyclic group [$M\_4(2)$](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/M4(2).html), of order $16$, has a **f**aithful **i**rreducible **c**omplex **r**epresentation (f.i.c.r.) of dimension $2$, whereas $\hat{\varphi}(M\_4(2)) = 0$.


Let $B$ be the subgroup generated by the minimal subgroups of $G$. By the Crosscut Theorem, if $H \in [1,G] \setminus [1,B]$ then $\mu(1,H) = 0$. Then $$\hat{\varphi}(G) = |G:B| \hat{\varphi}(B)$$ So that $\hat{\varphi}(G) \neq 0 $ if and only if $\hat{\varphi}(B) \neq 0$. But we can't expect that the existence of a f.i.c.r. for $G$ implies the existence of a f.i.c.r. for $B$ (whereas the converse is true).  

 In fact, $G = M\_2(4)$ has a f.i.c.r. but not its $B = C\_2^2$ 


*Improved question*: Is $(\Leftarrow)$ true if $B=G$?  

[note that $B$ is normal (see [here](https://math.stackexchange.com/a/2200591/84284)), so $B=G$ for $G$ simple] 


*Answer:* No, there are exactly two counter-examples of order $\le 100$: [$D\_8 \rtimes C\_2^2$](https://people.maths.bris.ac.uk/~matyd/GroupNames/61/D8sC2%5E2.html) and [$D\_8 \rtimes S\_3$](https://people.maths.bris.ac.uk/~matyd/GroupNames/73/D8sS3.html). They are of order $64$ and $96$ resp., and both check $B=G$, $\hat{\varphi}(G)=0$ and have a f.i.c.r. of dimension $4$. 


We can still specialized the question to $G$ simple.




---


For answering the comment of Benjamin Steinberg, we will prove an alternative result involving the normal subgroup lattice, and giving a formula shorter than in Pálfy's paper (available [here](https://drive.google.com/file/d/0B2P_JgZe-Zd0OG5LNnlrRG90Umc/view?usp=sharing)).


Let $G$ be a finite group and $\mathcal{N}(G)$ its normal subgroup lattice. Let $\mu\_{\mathcal{N}}$ be the Möbius function for $\mathcal{N}(G)$. Consider the dual normal Euler totient:
$$ \hat{\varphi}\_{\mathcal{N}}(G) = \sum\_{H \in \mathcal{N} (G)}\mu\_{\mathcal{N}}(1,H) |G:H| $$ Let $V\_1, \dots, V\_r$ be equivalent class representatives of the irreducible complex representations of $G$.


*Theorem*: $\hat{\varphi}\_{\mathcal{N}}(G) = \sum\_{V\_i \text{ faithful}} \dim(V\_i)^2$.  

*Corollary:* $G$ has a faithful irreducible complex representation iff $\hat{\varphi}\_{\mathcal{N}}(G) \neq 0$.




---


*Proof of the theorem:* 


Let the fixed-point subspace $V\_i^H=\{v \in V\_i \ | \ hv=v ,\ \forall h \in H \}$ 


*[Claim 1](https://math.stackexchange.com/q/1740091/84284)*: $|G:H| = \sum\_i\dim(V\_i) \dim(V\_i^H)$  

*Claim 2*: $H$ is a normal subgroup iff $\forall i \ V\_i^H = \{0\}$ or $V\_i$.  

*Claim 3*: $V\_i$ is not faithful iff there is an atom $H$ of $\mathcal{N}(G)$ such that $V\_i^H = V\_i$.


Consider the set $L=\{ \bigoplus\_i V\_i^H \ | \ H \in \mathcal{N}(G) \}$ whose lattice structure is the reverse of $\mathcal{N}(G)$.  

By applying the Crosscut Theorem and the inclusion-exclusion principle, together with the claims above, the result follows. $\square$ 


*Reference request*: does this theorem exist in the literature?


