  Motivated by the recent work of Zhi-Wei Sun on determinants involving the
Legendre symbol, in this paper, we study some matrices concerning subgroups of
finite fields.
  For example, let $q\equiv 3\pmod 4$ be an odd prime power and let $\phi$ be
the unique quadratic multiplicative character of the finite field
$\mathbb{F}_q$. If set $\{s_1,\cdots,s_{(q-1)/2}\}=\{x^2:\
x\in\mathbb{F}_q\setminus\{0\}\}$, then we prove that
  $$\det\left[t+\phi(s_i+s_j)+\phi(s_i-s_j)\right]_{1\le i,j\le
(q-1)/2}=\left(\frac{q-1}{2}t-1\right)q^{\frac{q-3}{4}}.$$
  This confirms a conjecture of Zhi-Wei Sun.
