This question came up in a discussion between me and a friend. We recently learned a bit about representation theory. We then realized that the regular symmetric matrices over any field form a group. I now wonder what kind of group this is and wether it is the representation of some otherwise known group.
I already realized that when using $\mathbb{F}_2$ symmetric matrices are always boring, if regular, as regular and symmetric is only satisfied by the identity matrix. Is this maybe true for all finite (prime) fields? I realized (from some experiments) it might be true for $\mathbb{F}_3$ as well. If so then what are interesting cases of symmetric matrix groups and what groups do those correspond to?
I am not only happy for some answers, but would also love to get some reading advice on this topic. Thank you everyone in advance.
Edit: To clarify two things that were asked. A matrix group is the representation of another group if it is isomorphic to this group. By regular I mean that the determinant is not 0.
Best Answer
The invertible symmetric matrices don't form a group (in the obvious way, that is, they're not closed under multiplication). If $A$ and $B$ are symmetric then $(AB)^T = B^T A^T = BA$ so $AB$ is symmetric iff $A, B$ commute.
You can try to fix this by symmetrizing the product to $A \circ B = \frac{AB + BA}{2}$ (assuming $2$ is invertible) but 1) this does not preserve invertible matrices and 2) it is also not associative. However, it does endow the vector space of symmetric matrices with the structure of a Jordan algebra.