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ON QUASI-REPRESENTING GRAPHS FOR A CLASS OF B(1)-GROUPS

  • Yom, Peter Dong-Jun
  • Received : 2010.05.30
  • Published : 2012.05.01

Abstract

In this article, we give a characterization theorem for a class of corank-1 Butler groups of the form $\mathcal{G}$($A_1$, ${\ldots}$, $A_n$). In particular, two groups $G$ and $H$ are quasi-isomorphic if and only if there is a label-preserving bijection ${\phi}$ from the edges of $T$ to the edges of $U$ such that $S$ is a circuit in T if and only if ${\phi}(S)$ is a circuit in $U$, where $T$, $U$ are quasi-representing graphs for $G$, $H$ respectively.

Keywords

Butler groups;$\mathcal{B}^{(1)}$-groups;quasi-representing graphs;quasi-isomorphisms

References

  1. D. Arnold and C. Vinsonhaler, Representing graphs for a class of torsion-free abelian groups, Abelian Group Theory (Oberwolfach, 1985), 309-332, Gordon and Breach, New York, 1987.
  2. D. Arnold and C. Vinsonhaler, Quasi-isomorphism invariants for a class of torsion-free abelian groups, Houston J. Math. 15 (1989), no. 3, 327-340.
  3. D. Arnold and C. Vinsonhaler, Invariants for a class of torsion-free abelian groups, Proc. Amer. Math. Soc. 105 (1989), no. 2, 293-300. https://doi.org/10.1090/S0002-9939-1989-0935102-X
  4. D. Arnold and C. Vinsonhaler, Duality and Invariants for Butler groups, Pacific J. Math. 148 (1991), no. 1, 1-9. https://doi.org/10.2140/pjm.1991.148.1
  5. F. Richman, An extension of the theory of completely decomposable torsion-free abelian groups, Trans. Amer. Math. Soc. 279 (1983), no. 1, 175-185. https://doi.org/10.1090/S0002-9947-1983-0704608-X
  6. P. Yom, A characterization of a class of Butler groups, Comm. Algebra 25 (1997), no. 12, 3721-3734. https://doi.org/10.1080/00927879708826080
  7. P. Yom, A characterization of a class of Butler groups II, Abelian group theory and related topics (Oberwolfach, 1993), 419-432, Contemp. Math., 171, Amer. Math. Soc., Providence, RI, 1994.
  8. P. Yom, A relationship between vertices and quasi-isomorphism for a class of bracket groups, J. Korean Math. Soc. 44 (2007), no. 6, 1197-1211. https://doi.org/10.4134/JKMS.2007.44.6.1197