Let $F,\,F'$ be two locally free sheaves (of rank $n$) over $\mathbb{P}^1_k$ for some field $k$ such that $F'$ is a subsheaf of $F$. Let $E'$ and $E$ denote the corresponding vector bundles.
The determinant bundle $E$ is the line bundle $ \det E$ defined by the gluing data determined by the determinant of the transition matrices of trivializations of $E$.
Now I wonder if $G,\,G'$ denote the locally free sheaves of rank 1 corresponding to $\det E,\,\det E'$, is $G'$ a subsheaf of $G$?
Another linked question is
Do we have at least $\deg \det E' \leq \deg \det E$?
My original thoughts were considering the invariants of $E,\,E'$ regarding their unique decomposition into a direct sum of line bundles $$E = \bigoplus_{i=1}^n \mathcal{O}_{\mathbb{P^1}}(d_i),\quad E' = \bigoplus_{i=1}^n \mathcal{O}_{\mathbb{P^1}}(d_i'). $$ I think, in general, we do not have a relation such $d_i' \leq d_i$ but I assume we have at least $\sum_{i=1}^n d_i' \leq \sum_{i=1}^n d_i$. But the latter two are nothing else than the degrees of $\det E'$ and $\det E$.
I would appreciate any kind of help, references etc. Thank you very much!
If $f \colon F' \to F$ is an embedding of sheaves of the same rank, then $\det(f) \colon \det(F') \to \det(F)$ is an embedding too.