Let $A$ be an abelian normal subgroup of a group $G$ and let {$g_i| 1\le i\le m$}, be a set of left coset representatives of $A$ in $G$. Then, for each choice of $i$ and $j$, we have $$g_ig_j=g_ra_{ij}$$ for some $r$ and $a_{ij}\in A$.
In the case that $m$ and $|A|$ are coprime, properties of the matrix $(a_{ij})$ and how it is transformed under a change of representatives provide a reasonably elegant way of writing out a proof that $A$ has a complement in $G$ (the Schur-Zassenhaus Theorem).
However, my question is - have properties of these matrices been studied in general i.e. when $m$ and $|A|$ are not necessarily coprime?
The following example shows that there might be properties worth investigating (if this work has not already been done).
For $(a_{ij})$ defined as above, the product of all the elements of $(a_{ij})$ is in $Z(G)$.
Proof
For any choice of $i,j,k$ let $g_ig_j\in g_rA$, $g_jg_k\in g_sA$, $g_ig_jg_k\in g_tA$. Then $$(g_ig_j)g_k= g_ra_{ij}g_k=g_rg_k(a_{ij})^{g_k}=g_ta_{rk}(a_{ij})^{g_k}$$ and also $$g_i(g_jg_k)= g_ig_sa_{jk}=g_ta_{is}a_{jk}.$$
Let $z$ denote the product of all the elements of $(a_{ij})$ and let $a(k)$ denote the product of all the elements in the $k$th column of $(a_{ij})$.
Equating the two expressions for $g_ig_jg_k$ gives
$$a_{rk}(a_{ij})^{g_k}=a_{is}a_{jk}$$ and therefore $$\prod_{i,j}a_{rk}(a_{ij})^{g_k}=\prod_{i,j}a_{is}a_{jk}.$$ As $i$ and $j$ run over all values from $1$ to $m$, $r$ runs over all values $m$ times and $s$ runs over all values once (independently of $i$). Therefore $$a(k)^mz^{g_k}=za(k)^m\implies z^{g_k}=z.$$
Then $<A,g_1, ..., g_m> \subseteq C(z)$ and so $z\in Z(G)$.