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For a group $G$, there are two elements a, b which are "relation-free", i.e., there is no nonempty, reduced word $W(X,Y)$ such that $W(a,b)=1$ in $G$.

Is there any terminologies or theories for that?

(My group is a free product of finite cyclic groups, and I'm considering one element is not contained in a factor, but is a product of generators of each factors.)

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    $\begingroup$ In other words, $\langle a,b \rangle$ is a free subgroup with $\{a,b\}$ as a basis, no? $\endgroup$
    – AGenevois
    Commented Aug 11, 2020 at 5:33
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    $\begingroup$ In addition to the comment of @AGenevois, as your setting is in the subgroups of a free product of finite cyclic groups you may be interested in the Kurosh subgroup theorem. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Aug 11, 2020 at 8:03
  • $\begingroup$ I'm calling this a "free pair", and more generally $(x_1,\dots,x_n)$ a free $n$-tuple if the corresponding map $F_n\to G$ is injective. $\endgroup$
    – YCor
    Commented Aug 12, 2020 at 5:36
  • $\begingroup$ @AGenevois;@Carl-Fredrik Nyberg Brodda Yes, it is. Maybe, I should ask more, I want to know some about a normally generated subgroup with the two elements. For instance, $\langle X^n, Y\rangle \subsetneq \langle X,Y\rangle$ for any $n>1$, but I don't know whether $\langle\langle X^n, Y\rangle\rangle_G \subsetneq \langle\langle X,Y \rangle\rangle_G$. $\endgroup$
    – qkqh
    Commented Aug 12, 2020 at 6:00
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    $\begingroup$ What you are asking in the comments is essentially unrelated to the question. You could ask a new one. I will say there is probably subtlty in what you are asking. For example, it is non trivial to show that $G$ isn't normally generated by a single element. $\endgroup$
    – user35370
    Commented Aug 13, 2020 at 18:45

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