Does there exist a group $G$ (finite or infinite) with three subgroups $A, B, C \leq G$ satisfying the following three conditions?



> 
> 1. $A = N\_G(A)$, $B = N\_G(B)$, $C = N\_G(C)$;
> 2. $AB = BC = CA = G$;
> 3. $A \cap B = B \cap C = C \cap A = 1$.
> 
> 
> 


(This question turned up in a more specific setting, but a negative answer to the existence of such a group would settle our case.)



Groups with triple system of self-normalizing subgroups
I revise earlier edits to give a coherent account of the construction which shows that such subgroups can exist. 


The underlying idea of the strategy is as follows: Let $X$ be a non-trivial finite group with trivial center which admits an automorphism $\alpha$ fixing only the identity (informally, and by a slight abuse, known as fixed-point free automorphism). Note that $\alpha$ can't have prime order, for otherwise $X$ would be nilpotent by Thompson's theorem, and hence would have non-trivial center. Then the direct product $X \times X$ is a product of two self normalizing subgroups of order $|X|$ which intersect trivially. One is $\Delta(X) = \{ (x,x): x \in X \}$. The other is 
$\Delta^{\alpha}(X) = \{ (x,x\alpha): x \in X\}$. That $\Delta(X)$ is self-normalizing is clear,
since $Z(X) = 1$. For notice that if $(x,x)^{(a,b)} \in \Delta(X)$ for each $x \in X$, then 
$ab^{-1} \in Z(X)$. The argument for the other subgroup is similar. Now we seek such a group $X$ with trivial center admitting a fixed point free automorphism $\alpha$ of composite odd order. If such a group exists, we may set $\beta = \alpha^{-1}$. Then the group $G = X \times X$ has the desired three self-normalizing subgroups $\Delta(X)$, $\Delta^{\alpha}(X)$ and $\Delta^{\beta}(X)$. For notice that if $x \alpha = x \beta$, then $\alpha^{2}$ fixes $x$, so that $\alpha$ fixes $x$ as $\alpha$ has odd order. But then $x$ is the identity by hypothesis. Hence any two of the three subgroups have trivial intersection, and have product $X \times X$ by order considerations.


There does exist a group $X$ of order $7^{6} .2^{4}$ which admits a fixed point free automorphism $\alpha$ of order $9$, and which has trivial center. Hence the construction above does work in this case, and $G = X \times X$ has three self-normalizing subgroups of the required form. We find a subgroup $Y$ of order $144$ of ${\rm GL}(6,7)$ which has an elementary Abelian normal subgroup $U$ of order $16$, acted on by an element $a$ of order $9$ whose cube centralizes $U$ but such that $a$ itself acts fixed-point freely on $U$, and such that, furthermore, $a$ does not have the eigenvalue $1$ in the given representation. We then take $X$ to be the semidirect product $VU$, where $V$ is a $6$-dimensional vector space over ${\rm GF}(7)$ and $U$ acts as the given elementary Abelian subgroup of ${\rm GL}(6,7)$. Then allowing $a$ to act on $U$ as it does within ${\rm GL}(6,7)$, and to act on $V$ as the given matrix yields an action of $a$ on $VU$ as a fixed-point free automorphism of order $9$. In terms of matrices, $a$ is the matrix
$ \left( \begin{array}{clcrc} 0 & 1 & 0 & 0 & 0 & 0\\ 0&0&1&0&0&0\\ 2 &0 &0 &0&0&0\\
0&0&0&0&1&0\\ 0&0&0&0&0 &1\\0 &0 &0 &4&0& 0 \end{array}\right)$. The group $U$
is the set of diagonal matrices with diagonal entries $\pm 1$ such that the product of the
first three diagonal entries is $1$ and the product of the last three diagonal entries is $1$.
The subgroup $U$ is $a$-invariant, $a^3$ centralizes $U$, but $C\_{U}(a) = I$. The group $VU\langle a \rangle$ is the semidirect product $VY$.
A similar construction works for other odd primes $p$ by considering ${\rm GL}(2p,q)$, where $q$
is a prime congruent to $1$ (mod p).


