Protasov, I. V. Maximal topologies on groups. (English. Russian original) Zbl 0935.22002 Sib. Math. J. 39, No. 6, 1184-1194 (1998); translation from Sib. Mat. Zh. 39, No. 6, 1368-1381 (1998). The article is devoted to the notion of a maximal topological group. A topological space without isolated points is called a maximal space if the space has an isolated point in every stronger topology. A topological group is called maximal if the underlying topological space of the group is a maximal space.The problem of the existence of a maximal topological group is a problem of set theory by nature. However, maximal homogeneous spaces and left-topological groups can be easily constructed in \(ZFC\).The author proves that, in \(ZFC\), an arbitrary infinite group can be endowed with a maximal regular left-invariant topology. In addition, the construction of this topology yields a solution to the Hindman-Strauss problem of a regular idempotent [Semigroup Forum 51, No. 3, 299-318 (1995; Zbl 0843.22005)]. Besides, the author proves some other theorems about groups with strongest left-invariant topology. Reviewer: K.N.Ponomarev (Novosibirsk) Cited in 2 ReviewsCited in 8 Documents MSC: 22A05 Structure of general topological groups 54H11 Topological groups (topological aspects) Keywords:topological group; maximal topological group; homogeneous space; left-topological group Citations:Zbl 0843.22005 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] V. I. Malykhin, ”Extremally disconnected and nearly extremally disconnected groups,” Dokl. Akad. Nauk,220, No. 1, 27–30 (1975). [2] I. V. Protasov, ”Filters and topologies on semigroups,” Mat. Studii. Pratsi L’vivsk. Mat. Tovarishch., No. 3, 15–28 (1994). · Zbl 0927.22009 [3] V. I. Malykhin, ”Extremally disconnected topological groups,” Uspekhi Mat. 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