Minimum simplicial complexes with given abelian automorphism group

Author:
Zevi Miller

Journal:
Trans. Amer. Math. Soc. **271** (1982), 689-718

MSC:
Primary 05C65; Secondary 20B25

DOI:
https://doi.org/10.1090/S0002-9947-1982-0654857-3

MathSciNet review:
654857

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Abstract: Let be a pure -dimensional simplicial complex. Let be the automorphism group of , and let be the group of permutations on -cells of induced by the elements of . Given an abelian group we consider the problem of finding the minimum number of points in such that , and the minimum number of -cells in such that . Write , where each factor appears times in the canonical factorization of . For containing no factors satisfying we find that when , and we derive upper bounds for and in the remaining possibilities for and .

**[1]**W. Arlinghaus,*The smallest graphs with given abelian group*, Thesis, Wayne State University, 1979.**[2]**László Babai,*On the minimum order of graphs with given group*, Canad. Math. Bull.**17**(1974), no. 4, 467–470. MR**0406855**, https://doi.org/10.4153/CMB-1974-082-9**[3]**A. K. Dewdney,*Extensions and generalizations of graph theorems to complexes and hypergraphs*, Thesis, University of Waterloo, 1974.**[4]**Frank Harary,*Graph theory*, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. MR**0256911****[5]**Richard A. Duke and Frank Harary,*Generalized Ramsey theory. VI. Ramsey numbers for small plexes*, J. Austral. Math. Soc. Ser. A**22**(1976), no. 4, 400–410. MR**0434863****[6]**Frank Harary and Ed Palmer,*The smallest graph whose group is cyclic*, Czechoslovak Math. J.**16 (91)**(1966), 70–71 (English, with Russian summary). MR**0194353****[7]**F. Harary and E. M. Palmer,*On the point-group and line-group of a graph*, Acta Math. Acad. Sci. Hungar.**19**(1968), 263–269. MR**0231753**, https://doi.org/10.1007/BF01894508**[8]**Donald J. McCarthy and Louis V. Quintas,*A stability theorem for minimum edge graphs with given abstract automorphism group*, Trans Amer. Math. Soc.**208**(1975), 27–39. MR**0369148**, https://doi.org/10.1090/S0002-9947-1975-0369148-4**[9]**Donald J. McCarthy and Louis V. Quintas,*The construction of minimal-line graphs with given automorphism group*, Topics in graph theory (New York, 1977) Ann. New York Acad. Sci., vol. 328, New York Acad. Sci., New York, 1979, pp. 144–156. MR**557894****[10]**R. L. Meriwether,*Smallest graphs with a given cyclic group*(unpublished).**[11]**Z. Miller,*Minimum simplicial complexes with given abelian automorphism group*, Thesis, University of Michigan, 1979.**[12]**Gert Sabidussi,*On the minimum order of graphs with given automorphism group*, Monatsh. Math.**63**(1959), 124–127. MR**0104596**, https://doi.org/10.1007/BF01299094**[13]**W. R. Scott,*Group theory*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0167513****[14]**I. M. Vinogradov,*Elements of number theory*, Dover Publications, Inc., New York, 1954. Translated by S. Kravetz. MR**0062138**

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0654857-3

Article copyright:
© Copyright 1982
American Mathematical Society