Kalmbach, Gudrun Automorphism groups of orthomodular lattices. (English) Zbl 0538.06009 Bull. Aust. Math. Soc. 29, 309-313 (1984). It is proved that every group is the automorphism group of some orthomodular lattice. This generalizes the result of G. Schrag who solved this problem in the case of finite groups [Proc. Am. Math. Soc. 55, 243-249 (1976; Zbl 0352.06007)]. The proof uses the following results: Birkhoff’s construction of a lattice with a given group as its automorphism group; the embedding of a lattice L in an orthomodular lattice generated by the chains in L; the existence of Boolean algebras with trivial automorphism groups; and the pasting of orthomodular lattices at atoms. Reviewer: G.A.Fraser Cited in 3 ReviewsCited in 9 Documents MSC: 06C15 Complemented lattices, orthocomplemented lattices and posets 08A35 Automorphisms and endomorphisms of algebraic structures 20F29 Representations of groups as automorphism groups of algebraic systems Keywords:automorphism group; Boolean algebras with trivial automorphism groups; pasting; orthomodular lattices Citations:Zbl 0352.06007 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Kalmbach, Orthomodular lattices (1983) [2] Jónsson, Topics in universal algebra 250 (1972) · doi:10.1007/BFb0058648 [3] Gudder, Canad. J. Math. 23 pp 659– (1971) · Zbl 0256.43012 · doi:10.4153/CJM-1971-073-1 [4] Greechie, J. Austral. Math. Soc. 25 pp 19– (1978) [5] Birkhoff, Lattice theory 25 (1973) [6] Kalmbach, Arch. Math. (Basel) 28 pp 7– (1977) · Zbl 0356.06017 · doi:10.1007/BF01223881 [7] Rosenstein, Linear orderings (1982) [8] Sabidussi, Canad. J. Math. 9 pp 515– (1957) · Zbl 0079.39202 · doi:10.4153/CJM-1957-060-7 [9] Morash, Canad. J. Math. 25 pp 261– (1973) · Zbl 0271.06008 · doi:10.4153/CJM-1973-026-2 [10] McKenzie, Infinite and finite sets 10 pp 951– (1973) [11] DOI: 10.2307/2041882 · Zbl 0352.06007 · doi:10.2307/2041882 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.