Do Sylow subgroups distinguish representations? Let $G$ be a finite group, $X$ the set of elementary subgroups of $G$, $Y$ the set of Sylow subgroups of $G$.
Propsition 29 in Serre's Linear Representations of Finite Groups implies that the set $X$ distinguish representations, in the sense that

For any finite dimensional complex G-representations $V$ and $W$ whose restrictions to $X_i$ are isomorphic for each element $X_i \in X$, $V$ and $W$ are isomorphic.



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*I wonder if $Y$ distinguishes representations as well? I have tried some small groups like $A_3, A_4, S_4..$. Note that when $G$ is a compact real Lie group, any maximal torus distinguishes. I'd hope to say Sylow $p$-subgroups (with all possible $p$) should have similar properties, thus asking this question. Thank you.


EDIT: Derek has shown by an explicit example that $Y$ is not enough below! Notice also that the elementary subgroups are necessary for $R(G) \to \oplus_{H\in\text{some-set}} R(H)$ to be one-to-one (Green's theorem). For me, this justify why the elementary subgroups are important.


*How about this: can $Y$ distinguish all irreducible representations? This is actually what I had in mind.. but I was not careful enough. If we want to disprove this statement, we have to find two nonisomorphic irreducible representations $V$ and $W$ of $G$ such that their restrictions to any Sylow subgroups are isomorphic.


EDIT: A counter example has been found by Derek: $G=D_{2\times 12}$! The faithful irreps of degree $2$ (there are only two of them) restrict to the same ones to the Sylow-$2$ and Sylow-$3$.

The set $Y$ of Sylow subgroups does not distinguish between complex representations.
Let $G$ be cyclic of order $6$, and consider the two representations $\rho_1$ and $\rho_2$ of $G$ of degree 2  that map a generator $g$  of $G$ to
$$ \left(\begin{array}{cc}\omega&0\\0&-\omega^2\end{array}\right)\ \ \ \ \mathrm{and}\ \ \ \ 
\left(\begin{array}{cc}-\omega&0\\0&\omega^2\end{array}\right),$$ where $\omega$ is a cube root of $1$.
By computing the actions on $g^2$ and $g^3$, you can check that the restirctions of $\rho_1$ and $\rho_2$ to Sylow $3$- and $2$-subgroups of $G$ have the same character and are therefore isomorphic.
For an example with irreducible representations, we can take the two faithful irreducible representations of the dihedral group of order $24$, which are of degree 2.
