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Note Introductory to the Study of Klein's Group of Order 168

Published online by Cambridge University Press:  24 October 2008

H. F. Baker
H. F. Baker
Affiliation:
St John's College
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Extract

The group of order 168 discovered by Klein, which it is now known can be generated by two operations E, of order 7, and ϑ, of order 2, which satisfy the relations E7 = 1, ϑ2 = 1, (Eϑ)3 = 1, (E4ϑ)4 = 1, has a vast literature. But for the most part each author pursues the matter from his own point of view; and it seems it may be useful to present a simplified approach to the theory which takes account of various possible aspects, in particular the geometrical. This is the object of the present note; for most of its contents I have found it necessary to do fresh work, so that the paper is by no means a transcript of what is already available.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 4 , October 1935 , pp. 468 - 481
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

* Dickson, Linear Groups (1901), 303; he refers to Dyck and to Burnside.

† In a product AB of two operations, I think of the one written to the right as being the first carried out. But this is not important here, since, representing the group by permutations of seven numbers 0, 1, …, 6, if (Eϑ)3 = 1, (E 4ϑ)4 = 1, then (ϑ′E)3 = 1, (ϑ′E)4ϑ4 = 1, wherein, if ϑ = (ab) (cd), then ϑ′ = (6 − a, 6 − b) (6 − c, 6 − d).

‡ Klein, , Math. Ann. 14 (1878), 428–71CrossRefGoogle Scholar [Ges. Math. Abh. 3, 90–136]Google Scholar. Klein remarks that the group is omitted by Jordan in his enumeration of the finite groups of linear ternary substitutions (Journal für Math. 84 (1878), 89–215)Google Scholar. See also Klein, , Sitzungsber. Erlangen, 10 (1878), 110–11Google Scholar and Math. Ann. 15 (1879), 25–82Google Scholar [Ges. Math. Abh. 2, 388, 390Google Scholar]. An exhaustive account of the relations with elliptic modular equations is in Klein-Fricke, , Modul-functionen, 1 (1890), 692 ff.Google Scholar

§ Weber, , Algebra, 2 (1896), 475Google Scholar, who refers to Kronecker,

* Cf. Klein, , Ges. Math. Abh. 3, 103Google Scholar ff., and Weber, , Algebra, 2 (1896)Google Scholar, Chap. xiv (pp. 433 ff.).

* Hurwitz, , Math. Ann. 41 (1893), 424.Google Scholar

* Hesse, , Journal für Math. 49 (1855), 279–332.Google Scholar

† The form is found in Klein-Fricke, , Modulfunctionen. 1, 725Google Scholar. The geometrical consequences are not developed there.

* If S, T are two matrices of the same order such that S n = 1, T n = 1, and (S, T) = S n−1 + S n−2T + S n−3T 2+…+T n−1, (T, S) = T n−1 + T n−2S+ … + S n−1, we have S = (S, T) T (S, T)−1 = (T, S)−1T (T, S) provided that the matrices (S, T), (T, S) have non-vanishing determinants.

* Baker, , Principles of Geometry, 5, 216.Google Scholar

* Kowalevsky, , Acta Math. 4 (1884), 406.Google Scholar

† Baker, , Multiply-Periodic Functions (1907), 270.Google Scholar

‡ See Klein, , Ges. Math. Abh. 3, 112.Google Scholar

§ Hurwitz, , Math. Ann. 26 (1886), 117CrossRefGoogle Scholar; see also Halphen, , Math. Ann. 24 (1884), 461.CrossRefGoogle Scholar

* Baker, , Proc. London Math. Soc. (1), 35 (1903), 370.Google Scholar

* Cf. Klein, , Ges. Math. Abh. 3, 98.Google Scholar

† Klein, , Ges. Math. Abh. 3, 118.Google Scholar

‡ Baker, , Proc. London Math. Soc. (1), 35 (1903), 347Google Scholar; also 34 (1902), 355.