  Motivated by Bownik and Speegle's result on linear independence of wavelet
Parseval frames, we consider affine systems (analogous to wavelet systems)
defined on a second countable, locally compact abelian group $G$, where the
translations are replaced by the action of a countable, discrete subgroup
$\Gamma$ of $ G$ acting as a group of unitary operators on $L^2(G)$. The
dilation operation in the wavelet setting is replaced by integer powers of a
unitary operator $\delta$ onto $L^2(G)$. We show that, under some compatibility
conditions between $\delta$ and the action of the group $\Gamma$, the linear
independence of the translates of any function in $L^2(G)$ by elements of
$\Gamma$ implies the linear independence of affine Parseval frames in $L^2(G)$.
