Is every $n \times n$ matrix with entries in $\mathbb{Z}\_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL\_n(\mathbb{Z}\_p)$?


On the one hand, I know the analogous fact is false for matrices over $\mathbb{Z}$ with counterexamples constructed via the Latimer-MacDuffee theorem (but the only counterexamples I've seen break down in $\mathbb{Z}\_p$ for all $p$.) 


On the other hand I know there are various conditions under which $GL\_n(\mathbb{Q}\_p)$-conjugacy implies $GL\_n(\mathbb{Z}\_p)$-conjugacy, as well as lifting criteria from $GL\_n(\mathbb{F}\_p)$, so it seems possible that it could hold here.



Is a matrix similar to its transpose over $\mathbb{Z}_p$?
No for $n\geq 3$.


If $A\in M\_n(\mathbf Z\_p)$ were similar to $A^T\in M\_n(\mathbf Z\_p)$, then going modulo $p^2$, its image in $M\_n(\mathbf Z/p^2\mathbf Z)$ would be similar to the image of its transpose.


However, Pooja Singla, Steven Spallone and I have shown in [Similarity of matrices over local rings of length two](https://www.imsc.res.in/~amri/summaries.html#sim) that this fails for $n\geq 3$ (Theorem 7.9). For $n=3$ and $4$ we provide a classification and exact counts for matrices where this fails (see Table 9.4/ Table 8 in the arXiv version).


The answer is yes for $n=2$. This follows from Theorem 2.2 of [Similarity classes of 3x3 matrices over a local principal ideal ring](https://www.imsc.res.in/~amri/summaries.html#aopv). I thank @KConrad for nudging me to think this through.


