# Do $F$-traces of simple modules at $p'$-classes uniquely determine the module?

Let $$G$$ be a finite group and let $$p$$ be a prime number such that $$p\mid |G|$$.

Let $$\text{IBr}(G)$$ denote the set of irreducible Brauer characters of $$G$$ for the prime $$p$$.

Assume $$\mathbb{F}_{q}$$ is a splitting field for $$G$$ where $$q=p^f$$ for some positive integer $$f$$.

Set $$r:=|\text{IBr}(G)|$$.

Let $$\{\rho_1, ..., \rho_r\}$$ be the a set of representatives of all simple $$\mathbb{F}_{q}G$$-modules up to isomorphism.

It is well-known that $$\text{IBr}(G)$$ is linearly independent.

Now, take the $$\mathbb{F}_{q}G$$-traces of $$\rho_j$$ at the $$p'$$-classes and write the results in a vector, for each $$j$$.

Does the list of vectors obtained in that way always have the property that there are no repeated rows?

Example:

Doing the computations for $$G=A_5$$, the alternating group acting on $$5$$ symbols, for the prime number $$p=2$$ yields the following:

$$[ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ]$$

$$[ 0*Z(2), Z(2)^0, Z(2^2)^2, Z(2^2) ]$$

$$[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2 ]$$

$$[ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ]$$

The list of vectors obtained in that way has the property that no two rows are identical.

Is this always the case (for any finite group G and for any prime number p)?

A reference would be cool.

It is still the case that the $$\mathbb{F}_{q}$$-valued trace functions of the (say) $$\ell$$ non-isomorphic simple $$\mathbb{F}_{q}$$-modules $$V_{1},V_{2}, \ldots V_{\ell}$$, are linearly independent, where $$G$$ has $$\ell$$ $$p$$-regular classes.
The usual proof is to note that for each $$i$$ we may take an $$\mathbb{F}_{q}$$-linear combination of group elements which is represented by an idempotent of trace $$1$$ in $${\rm End}_{\mathbb{F}_{q}}(V_{i})$$ and represented by the zero matrix in $${\rm End}_{\mathbb{F}_{q}}(V_{j})$$ for all $$j \neq i$$.
Since (over $$\mathbb{F}_{q}$$) the trace of $$g$$ in its action on any $$V_{k}$$ is the trace of its $$p^{\prime}$$-part $$g_{p^{\prime}}$$, the claimed result follows.