Does there sometimes exist a nontrivial symmetry group of a linear system of equations? Take the symmetric looking linear system given by:
$$
A=\begin{pmatrix}
1&1&0&0&0&0&0\\
0&1&1&1&0&0&0\\
0&0&0&1&1&1&0\\
0&0&0&0&0&1&1
\end{pmatrix},\\
Ax=b
$$
Say I want to maximize $-\sum_i x_i$.   Then can we take advantage of the fact that rows 2&3 are essentially the same thing as well as rows 1&4 and thus reduce the burden of the solver using group theory as is done in many modern algorithms?   Note that in my application $x$ is itself a $\{0,1\}$-valued vector. And $b =(1,…,1)^T$.
What I mean by essentially the same thing is if $x_1 = x_2 = 1$ (the rest of $x$'s entries $0$) is a solution then, by symmetry of the cost function which is just a sum of $x$'s components, we have that $x_6 = x_7 = 1$ (the rest $0$) is also a solution and vise versa.  So the symmetry their could be something like $(1,6)(2,7)$.

Yes, such symmetry can be exploited via a technique called LP folding.  See https://arxiv.org/abs/1307.5697
