**Question 1 (the weak and simple statement, which, I think, already is wrong):** Let $p$ be a prime. Let $k$ be a field with characteristic $p$.

For any $k$-vector space $V$, consider the canonical projection $V^{\otimes p}\to \mathrm{Sym}^p V$ from the $p$-th tensor power of $V$ to the $p$-th symmetric power of $V$. This projection does not canonically split. But it induces a projection

$V^{\otimes p} / \left< v\otimes v\otimes ...\otimes v \text{ (}p\text{ times)} \mid v\in V\right> \to \mathrm{Sym}^p V / \left< vv...v \text{ (}p\text{ times)} \mid v\in V\right>$

(where the angular brackets mean "$k$-linear span"). Does this projection canonically split?

It trivially does for $p=2$, because it is the identity map in this case. But I highly suspect that it fails even for $p=3$. However I cannot prove it. Judging from Torsten Ekedahl's disproof of my previous guess, I should learn some modular representation theory of $\mathrm{GL}$; is there a good source for it?

If Question 1 actually happens to get a positive answer, then here is the question I am coming from:

**Question 2 (generalization of Question 1; ignore if Question 1 is answered No):** Much of the following is copied over from Restricted universal enveloping algebra of Abelian p-Lie algebra .

Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.

Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-module along with a $\mathbb Z$-linear map ${}^{[p]}:\mathfrak g\to\mathfrak g$ (written postfix) that satisfies $\left(\lambda v\right)^{[p]}=\lambda^p v^{[p]}$ for all $\lambda\in k$ and $v\in\mathfrak g$.

Let $U^{[p]}\left(\mathfrak g\right)$ be the restricted universal enveloping algebra of $\mathfrak g$. In other words, let $U^{[p]}\left(\mathfrak g\right)$ be the factor algebra of the symmetric algebra of $\mathfrak g$ modulo the ideal generated by elements of the form $x^p-x^{[p]}$ with $x\in\mathfrak g$. Note that $U^{[p]}\left(\mathfrak g\right)$ is not a graded algebra, but a filtered one.

Let $\otimes^{[p]}\left(\mathfrak g\right)$ be the factor algebra of the algebra $\otimes \mathfrak g$ (this is the tensor algebra of the $k$-module $\mathfrak g$) modulo the ideal generated by elements of the form $\underbrace{x\otimes x\otimes ...\otimes x}_{p\text{ times}}-x^{[p]}$ with $x\in\mathfrak g$.

The canonical projection $\otimes \mathfrak g\to\mathrm{Sym}\mathfrak g \to U^{[p]}\left(\mathfrak g\right)$ (where $\otimes \mathfrak g$ means the tensor algebra of $\mathfrak g$) induces a canonical projection

$\otimes^{[p]} \mathfrak g\to\mathrm{Sym}\mathfrak g \to U^{[p]}\left(\mathfrak g\right)$.

Does this projection split canonically?