All Questions

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 ...

I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that $-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ ...