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Let $A$ be a non-empty set and let $F(A)$ be the free group it generates. An element of $F(A)$ is of the form

$$w = a_1^{\varepsilon_1}a_2^{\varepsilon_2}\cdots a_{n-1}^{\varepsilon_{n-1}}a_n^{\varepsilon_n}$$

where $a_i\in A$ and $\varepsilon_i = \pm 1$ for all $i=1,\ldots,n$.

Let us fix a specific element, $b\in A$. We can define the group morphism $w_b:F(A)\to \mathbb{Z}$ where

$$w_b(a_i) = \begin{cases}1 & \text{if }a_i = b\\ 0 & \text{otherwise.} \end{cases}$$

With this definition,

$$w_b(w) = \sum_{\substack{1\leq i\leq n \\ a_i = b}} \varepsilon_i.$$

For example, $w_b(a^2b^{-1}cb^3a^{-1}) = 2$.

It seems like such a morphism would have a name in the literature. If so, what is it?

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  • $\begingroup$ It's a projection from the abelianization map from the free group on $A$ to the free abelian group on $A$. The abelianization map encodes $w_b$ for all $b$ simultaneously. $\endgroup$ Jan 8, 2018 at 23:43

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If $B$ is a subset of $A$, I would call projection from $F(A)$ onto $F(B)$ the morphism $\pi_B: F(A) \to F(B)$ defined, for each $a \in A$, by $$ \pi_B(a) = \begin{cases} a &\text{if $a \in B$}\\ 1 &\text{otherwise} \end{cases} $$ In particular, your morphism would be the projection from $F(A)$ onto $F(b)$.

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  • $\begingroup$ That's interesting, I see it more as the projection$F(A) \to F(A)/N(A\setminus\{b\}) \simeq F(\{b\}) \simeq \mathbb{Z}$ where $N(S)$ is the normal clausure of the set $S$. $\endgroup$
    – Darth Geek
    Jan 9, 2018 at 0:00

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