Power-conjugate equations in symmetric groups

Abstract

First we consider the solutions of the general “cubic” equation ${\alpha }_1°x^{r_1}°{\alpha }_2°x^{r_2}°{\alpha }_3°x^{r_3}=1$ (with $r_1,r_2,r_3\in \{1,-1\}$) in the symmetric group ${\mathrm{S}}_n$. In certain cases this equation can be rewritten as $\alpha °y°{\alpha }^{-1}=y^2$ or as $\alpha °y°{\alpha }^{-1}=y^{-2}$, where $\alpha \in {\mathrm{S}}_n$ depends on the α_i’s and the new unknown permutation $y\in {\mathrm{S}}_n$ is a product of x (or $x^{-1}$) and one of the permutations ${\alpha }_i^{\pm 1}$. Using combinatorial arguments and some basic number theoretical facts, we obtain results about the solutions of the so-called power-conjugate equation $\alpha °y°{\alpha }^{-1}=y^e$ in ${\mathrm{S}}_n$, where $e\in \mathbb{Z}$ is an integer exponent. A divisibility condition involving the type of α provide solutions with $\alpha °y\ne y°\alpha$ and a further condition gives the complete list of solutions. Some other divisibility assumptions concerning the type of α ensure that the solutions of $\alpha °y°{\alpha }^{-1}=y^e$ are exactly the solutions of $y^{e-1}=1$ in the centralizer of α. Slightly stronger assumptions provide a complete answer to the question, when our equation has only the trivial solution y = 1.

Keywords:

• Equations in groups
• symmetric group
• the conjugate equation
• the cycles and the type of a permutation

2020 Mathematics Subject Classification:

• 05A05
• 05E16
• 20B30
• 20B05
• 20E45
• 20F70

Acknowledgments

The authors would like to sincerely thank the several insightful comments and suggestions of the referee.

Additional information

Funding

The second named author was partially supported by the National Research, Development and Innovation Office of Hungary (NKFIH) K138828.