I am writing a paper to introduce group theory through symmetry in molecules and group classification. How can (adjacency?) matrices represent molecules (abstract structure) and be used (through certain matrix operations) to deduce what symmetry/point group this molecule (or abstract structure) belongs to?
Also, by interest (as a small extension) I am wondering whether adjacency matrices are applied in other areas than pure graph theory- do you have examples that I could investigate? Thanks!
The isomorphism classes of finitely generated Abelian groups correspond to normal forms of certain integer matrices. Moreover, this correspondence can be used to derive the Structure Theorem for Finitely Generated Abelian Groups.
Every finitely generated Abelian group can be described by a presentation matrix. This is an $n \times m$ integer matrix $A$ describing the group in terms of generators and relations. More precisely, the generators $\mathbf{v} = (v_1, \ldots, v_n)$ satisfy the equation
$$\mathbf{v} A = 0$$
For every integer matrix $A$, the Smith Normal Form of $A$ is the unique diagonal matrix with non-negative entries
$$D = \text{diag}(d_1, \ldots, d_k)$$
(padded with zeros so that it is $n \times m$) such that each $d_i \not= 0$ for $i \leq \ell$ and $d_i \mid d_{i + 1}$ for $i \leq \ell$ and
$$U A V = D$$
for some invertible integer matrices $U$ and $V$.
It can be shown that the isomorphism class of a finitely generated Abelian group corresponds to the Smith Normal Form of any presentation matrix of the group. In fact, this is equivalent to the Structure Theorem for Finitely Generated Abelian groups and is one way of proving that theorem.