Geodesic
Any group is a maximal subgroup of the semigroup of binary relations on some set†
Glasgow mathematical journal, Tome 14 (1973) no. 1, pp. 21-24

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We show that the theorem stated in the title is a corollary to a result of K. A. Zaretskii [5] and a theorem of G. Birkhoff [1]. The construction we use further shows that all groups with cardinal less than or equal to the cardinal of the given group are simultaneously realised as maximal subgroups of the same semigroup of binary relations Bx. For finite or countable groups, when Xmay be taken to be finite or countable, respectively, and for an entirely different method of proof, the paper of J. S. Montague and R. J. Plemmons [3] should be consulted. For two further proofs of the theorem of the title to this note, this time for any X, see also R. J. Plemmons and B. M. Schein [4] and A. H. Clifford [2]. 
Preston, G. B. Any group is a maximal subgroup of the semigroup of binary relations on some set†. Glasgow mathematical journal, Tome 14 (1973) no. 1, pp. 21-24. doi: 10.1017/S0017089500001683
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[1] 1.Birkhoff, Garrett, Sobre los grupos de automorfismos, Revista de la Union Mat. Argentina 9 (1946), 155–157. Google Scholar

[2] 2.Clifford, A. H., A proof of the Montague-Plemmons-Schein theorem on maximal subgroups of the semigroup of binary relations, Semigroup Forum 1 (1970/), 272–275. Google Scholar | DOI

[3] 3.Montague, J. S. and Plemmons, R. J., Maximal subgroups of the semigroup of relations, J. of Algebra 13 (1969), 575–587. Google Scholar | DOI

[4] 4.Plemmons, R. J. and Schein, B. M., Groups of binary relations, Semigroup Forum 1 (1970), 267–271. Google Scholar | DOI

[5] 5.əapeцkий, K. A. (K. A. Zaretskii), Пoлyrpyллa бинapныx οτношений (The semigroup of binary relations), Mat. Sb. 61 (1963), 291–305. Google Scholar

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