Journal of Applied and Pure Mathematics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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ITERATED MATRIX SEQUENCES MODULO 5 AND THEIR STABLE, AMICABLE, AND SOCIABLE STRUCTURES

  • JISU YANG (Department of Mathematics, Jeonbuk National University) ;
  • CHI YOUNG AHN (Division of Industrial Mathematics, National Institute for Mathematical Sciences) ;
  • DAEYEOUL KIM (Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University)
  • Received : 2025.11.19
  • Accepted : 2026.02.09
  • Published : 2026.03.30
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Abstract

This paper investigates the iterative behaviour of 2×2 matrices over the finite ring ℤ5 and relates it to the 5-torsion subgroup E[5] of an elliptic curve. Motivated by the work of Fan-Ahn-Kim [1] in the modulo-4 setting, we examine all 625 matrices in M2(ℤ5) and classify their iterated sequences into stable, amicable, and sociable types. For each matrix we determine the order, period, and perfect period, and we introduce the local entry-sum Ts and the global type Ty to encode the resulting orbits as periodic words over ℤ5. We also study the induced actions on the group Y = ℤ5 × ℤ5, which we regard as the 5-torsion subgroup E[5] of an elliptic curve. The corresponding subgroup sequences on Y are classified according to stability, amicability, and sociability, and several uniformity phenomena explained by the natural GL2(𝔽5)-symmetry are observed. In particular, some of the stable cases exhibit clover-like patterns among the six order-5 subgroups of Y . Explicit examples of elliptic curves E/𝔽31 satisfying E(𝔽31) ≅ ℤ5 × ℤ5 are given, and the matrix dynamics is illustrated by a simple color representation of all 625 matrices. Taken together, these results provide a concrete description of matrix iteration modulo 5 and its connection with the endomorphism structure of elliptic curves.

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Acknowledgement

This work was supported by the National Institute for Mathematical Sciences (NIMS) grant funded by the Korea government (No. B26A10000). And this study was produced based on the master's thesis of Jisu Yang.

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