If we let $r$ be a natural number, $\lambda$ be a partition of $r$, $\Sigma_r$ be the symmetric group on $r$ numbers, we can define the following $K\left[ \Sigma_r \right]$-module:
$M^\lambda := \operatorname{span}_K \{\text{tabloids of shape } \lambda\}$. See for example G D James' Representation Theory of the Symmetric Group.
I've also seen that the module $M^\lambda$ can be defined as the induced representation of the trivial module over the Young Subgroup $\Sigma_\lambda := \Sigma_{\lambda_1} \times \Sigma_{\lambda_2} \times \ldots$ , for $\lambda = (\lambda_1,\lambda_2,\ldots)$, to $\Sigma_r$ i.e.:
$$M^\lambda = \operatorname{Ind}_{\Sigma_\lambda}^{\Sigma_r}(1),$$
where 1 denotes the trivial $K\left[\Sigma_\lambda\right]$-module.
I'm trying to prove that these two definitions give isomorphic modules, by finding a direct isomorphism or by proving that they are equivalent representations, but can't seem to generelize the basic examples I'm looking at.
If anyone can help, I'd appreciate it. Note: I'm using the coset defintiion of the induced module, not the definition as the tensor product.