Showing $V\cong W$ if $\dim V^H=\dim W^H$ I am trying to show that if $W$ and $V$ are to $\mathbb{Q}[G]$ modules then $V\cong W$ if $\dim V^H=\dim W^H$ for all cyclic $H\leq G$ ( where $V^H$ denotes the invariant subspace under $H$
So I have this as a corollary to Artin theorem so I thought I would proceed in the following way:
$\dim V^H=[V\downarrow_H, \mathbb{I}_H]_H=\sum_{k\leq G, K cyclic } \frac{a_K}{[N_G(K):K]}[\mathbb{I}_K\uparrow^G\downarrow_H,\mathbb{I}_H]_H$ 
Now I want to apply Mackey to  this and I will get:
$\sum_{k\leq G, K cyclic } \frac{a_K}{[N_G(K):K]}[\sum_{g\in H\ G/K} \mathbb{I}\downarrow_{H\cap g K g^{-1}}\uparrow^{H},\mathbb{I}_H]_H$
Then applying Frobenius to get:
$\sum_{k\leq G, K cyclic } \frac{a_K}{[N_G(K):K]}\sum_{g\in H\ G/K} [\mathbb{I}\downarrow_{H\cap g K g^{-1}},\mathbb{I}_{H\cap gKg^{-1}}]_{H\cap gKg^{-1}}$
But then is $[\mathbb{I}\downarrow_{H\cap g K g^{-1}},\mathbb{I}_{H\cap gKg^{-1}}]_{H\cap gKg^{-1}}=1$ so this doesn't really seem to give me anything?
 A: Here's a sketch (incomplete) of the way I would do it: for a give $g \in G,$ let $\theta_{g}$ be the class function of $G$ which takes value $|G|$ at $x$ if $x$ is $G$-conjugate to a generator of $\langle g \rangle$. Then $\theta_{g}$ is a generalized character of $G,$ and is in fact a $\mathbb{Z}$-combination of permutation characters induced from subgroups of $\langle g \rangle.$ This is essentially one way that Artin's induction theorem is proved.
Let $\alpha$ be the character of $G$ afforded by $V$ and $\beta$ be the character f $G$ afforded by $W.$ Then the hypotheses, and Frobenius reciprocity, give that $\langle \theta_{g},\alpha \rangle = \langle \theta_{g}, \beta \rangle$ for each $g \in G.$ But $\alpha$ and $\beta$ take respective constant values $\alpha(g)$ and $\beta(g)$ where $\theta_{g}$ does not vanish, so we obtain $\alpha(g) = \beta(g)$ for all $g \in G.$ Hence $V$ and $W$ afford the same character so are isomorphic (this takes a little "descent" argument from the isomorphism as $\mathbb{C}G$-modules).
