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YCor
YCor
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I will pose the question in the form in which it originally appeared to me:

Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\mathcal{Z}$. Consider the words $QabRcd$ and $QbaRdc$ as being cyclic, e.g. by placing them on separate circles.

Do there existsexist different letters $a,b,c,d$ and words $Q,R$ such that that the cyclic word $QabRcd$ can be obtained from $QbaRdc$ by rotation?

The appropriate context for this questions seems to be whether this equation has a solution in a conjugacy class of the free group generated by the letters of $\mathcal{Z}$, and hence the title.

I will pose the question in the form in which it originally appeared to me:

Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\mathcal{Z}$. Consider the words $QabRcd$ and $QbaRdc$ as being cyclic, e.g. by placing them on separate circles.

Do there exists different letters $a,b,c,d$ and words $Q,R$ such that that the cyclic word $QabRcd$ can be obtained from $QbaRdc$ by rotation?

The appropriate context for this questions seems to be whether this equation has a solution in a conjugacy class of the free group generated by the letters of $\mathcal{Z}$, and hence the title.

I will pose the question in the form in which it originally appeared to me:

Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\mathcal{Z}$. Consider the words $QabRcd$ and $QbaRdc$ as being cyclic, e.g. by placing them on separate circles.

Do there exist different letters $a,b,c,d$ and words $Q,R$ such that the cyclic word $QabRcd$ can be obtained from $QbaRdc$ by rotation?

The appropriate context for this questions seems to be whether this equation has a solution in a conjugacy class of the free group generated by the letters of $\mathcal{Z}$, and hence the title.

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Leon Staresinic
Leon Staresinic
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Equation in the conjugacy class of a free group

I will pose the question in the form in which it originally appeared to me:

Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\mathcal{Z}$. Consider the words $QabRcd$ and $QbaRdc$ as being cyclic, e.g. by placing them on separate circles.

Do there exists different letters $a,b,c,d$ and words $Q,R$ such that that the cyclic word $QabRcd$ can be obtained from $QbaRdc$ by rotation?

The appropriate context for this questions seems to be whether this equation has a solution in a conjugacy class of the free group generated by the letters of $\mathcal{Z}$, and hence the title.