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For any $n$, we describe all endomorphisms of the braid group $B_n$ and ofits commutator subgroup $B'_n$, as well as all homomorphisms $B'_n\to B_n$.These results are new only for small $n$ because endomorphisms of $B_n$ arealready described by Castel for $n\ge 6$, and homomorphisms $B'_n\to B_n$ andendomorphisms of $B'_n$ are already described by Kordek and Margalit for $n\ge7$. We use very different approaches for $n=4$ and for $n\ge 5$.

Homomorphisms of commutator subgroups of braid groups with small number of strings