Notes

¹It is often convenient to use digits rather than letters for the purpose of illustration.

²We shall sometimes use the phrase that two groups are of the same type to denote that they are simply isomorphic.

³Dyck, “Gruppentheoretische Studien,” Math. Ann., Vol. XX (1882), p. 30.

⁴“Algebraische Gebilde mit eindeutigen Transformationen in sich,” Math. Ann., Vol. XLI (1893), p. 421.

⁵Among these $T$ of course occurs.

⁶Among these sub-groups $H$ itself occurs.

⁷Strictly speaking, this statement should be qualiﬁed by the addition “except that formed by the identical operation alone.” No real ambiguity however will be introduced by always leaving this exception unexpressed.

⁸“Zur Reduction der algebraischen Gleichungen,” Math. Ann. XXXIV (1889), p. 31.

⁹Herr Frobenius has extended the use of the symbol to the case in which $Γ$ is any group, whether contained in $G$ or not, with which every operation of $G$ is permutable: “Ueber endliche Gruppen,” Berliner Sitzungsberichte, 1895, p. 169. We shall always use the symbol in the sense deﬁned in the text.

¹⁰Dyck, “Gruppentheoretische Studien,” Math. Ann. Vol. XXII (1883), p. 97.

¹¹On Abelian groups, the reader may consult Frobenius and Stickelberger, “Ueber Gruppen vertauschbarer Elemente,” Crelle’s Journal, Vol. LXXXVI (1879), p. 217; and a very complete discussion in the second volume of Herr Weber’s recently published Lehrbuch der Algebra. In the proof of the existence of a set of independent generating operations (§ 41) we have directly followed Herr Weber. The name “Abelian group” has been applied by M. Jordan (Traité des substitutions etc. pp. 171 et seq.) to an entirely diﬀerent class of groups, whose operations are not permutable. Most writers, we believe, have used the phrase in the sense deﬁned in the text. The connection of Abel’s name with groups of permutable operations is due to his having been the ﬁrst to investigate, with complete generality, the application of such groups to the theory of equations, “Mémoire sur une classe particulière d’équations résolubles algébriquement,” Crelle’s Journal, Vol. IV (1829), p. 131; or Collected Works, 1881 edition, Vol. I, p. 478.

¹²Sylow, Math. Ann. V. (1872), p. 588.

¹³Let $P$ be an operation or sub-group of $G$ , and let $H$ be a sub-group of $G$ which contains $P$ . If every operation of $H$ is permutable with $P$ , we shall speak of $P$ as “self-conjugate in $H$ .” The phrase is to be regarded as emphasizing the relation of $P$ to a sub-group of $G$ which contains it; and not as implying that $H$ is the greatest, or the only, sub-group of $G$ , within which $P$ is a self-conjugate operation or sub-group.

¹⁴Young, American Journal, vol. XV (1893), p. 132.

¹⁵Frobenius, “Ueber endliche Gruppen,” Berliner Sitzungsberichte (1895), p. 173: Burnside, “Notes on the theory of groups of ﬁnite order,” Proc. London Mathematical Society, Vol. XXVI (1895), p. 209.

¹⁶Young, “On groups whose order is the power of a prime,” American Journal of Mathematics, Vol. XV (1893), p. 171.

¹⁷Let $\begin{matrix} S_{1}^{α_{1}} = 1, S_{2}^{α_{2}} = 1, \dots, S_{n}^{α_{n}} = 1, \\ f_{1} (S_{i}) = 1, f_{2} (S_{i}) = 1, \dots, f_{k} (S_{i}) = 1, \end{matrix}$

where $f_{j} (S_{i})$ is an abbreviation for an expression of the form $S_{p}^{a} S_{q}^{b} \dots S_{r}^{c}$ , be a set of relations, such as was considered in § 18, which completely specify a group. We may then, without altering the sense in which the word “type” has been used (§§ 19, 44), speak of the type of group deﬁned by these relations. It is essential however that the relations should completely specify the group, as otherwise they will deﬁne more than one type. For instance, it is clear, from § 56, that the equations

Q^{p} = 1, P^{p^{2}} = 1, Q^{- 1} P Q = P^{α},

where

p

is a given prime, but

α

is not given, will deﬁne more than one type of group. Any data in fact, which completely specify a group, may be said to deﬁne a type of group. Thus in dealing with Abelian groups of order

p^{m}

, a type is deﬁned by a partition of

m

¹⁸Frobenius, “Verallgemeinerung des Sylow’schen Satzes,” Berliner Sitzungsberichte, (1895), p. 989.

¹⁹In each of the ﬁve distinct cases to which we have been led in the discussion contained in §§ 65–67, we have arrived at a set of deﬁning relations, containing no indeterminate symbols, such that in each case a set of generating operations can be chosen to satisfy these relations. To justify the statement, in each particular case, that such a set of relations gives a distinct type of group, it is ﬁnally necessary to verify that the relations actually deﬁne a group of order $p^{m}$ . In the cases dealt with in the text, this veriﬁcation is implicitly contained in the process by which the relations have been arrived at. We have therefore omitted the direct veriﬁcation, which moreover is extremely simple. We shall similarly omit the corresponding veriﬁcations in the discussion of groups of orders $p^{3}$ and $p^{4}$ , as in none of these cases does it present any diﬃculty. To illustrate the necessity of such a veriﬁcation in general, we may consider a simple case. The relations

P^{3} = 1, Q^{9} = 1, P^{- 1} Q P = Q^{α},

where

α

is any given integer, certainly deﬁne a group whose order is equal to or is a factor of

27

, since they indicate that

{P}

and

{Q}

are permutable. They will not however give a type of group of order

27

, unless

α

1

4

7

. For instance, if

α = 5

, the relations involve

Q = P^{- 3} Q P^{3} = Q^{8}, or Q^{7} = 1 .

Hence

Q = Q^{4 \cdot 7 - 3 \cdot 9} = 1,

and the relations hold only for a group of order

3

. Again, if

α = 3

, the relations give

P^{- 1} Q^{3} P = Q^{9} = 1, or Q^{3} = 1,

and as before they deﬁne a group of order

3

²⁰On groups of orders $p^{3}$ and $p^{4}$ , the reader may consult, in addition to Young’s memoir already referred to, Hölder, “Die Gruppen der Ordnungen $p^{3}$ , $p q^{2}$ , $p q r$ , $p^{4}$ ,” Math. Ann., XLIII, (1893), in particular, pp. 371–410.

²¹Cauchy, Exercises d’analyse, III, (1844), p. 250.

²²Sylow, Théorèmes sur les groupes de substitutions, Math. Ann., V (1872) pp. 584 et seq. Compare also Frobenius, Neuer Beweis des Sylow’schen Satzes, Crelle, C. (1886), p. 179.

²³Frobenius, “Ueber endliche Gruppen,” Berliner Sitzungsberichte (1895), p. 176: and Burnside, “Notes on the theory of groups of ﬁnite order,” Proc. London Mathematical Society, Vol. XXVI (1895), p. 209.

²⁴It should be noticed that, if $α$ is not a root of the congruence, the relations deﬁne a group of order $p_{1}$ . Thus from

P_{1}^{- 1} P_{2} P_{1} = P_{2}^{α},

we have

P_{1}^{- p_{1}} P_{2} P_{1}^{p_{1}} = P_{2}^{α^{p_{1}}} .

Hence

P_{2}^{α^{p_{1} - 1}} = 1, and P_{2}^{p_{2}} = 1,

so that

P_{2} = 1

, and the group reduces to

{P_{1}}

²⁵Miller, Comptes Rendus, CXXII (1896), p. 370.

²⁶Frobenius: Berliner Sitzungsberichte (1895), p. 988.

²⁷“Verallgemeinerung des Sylow’schen Satzes,” Berliner Sitzungsberichte (1895), pp. 984, 985.

²⁸Frobenius, Berliner Sitzungsberichte, (1895), p. 987.

²⁹Frobenius: “Ueber endliche Gruppen,” Berliner Sitzungsberichte (1895), p. 170.

³⁰Frobenius, loc. cit. p. 170.

³¹“Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen,” Math. Ann. XXXIV, (1889), p. 33.

³²“Traité des substitutions,” (1870), p. 42.

³³The reader will notice that $B$ can be eliminated from these relations, and that the group can be deﬁned by $A^{2} = 1$ , $R^{3} = 1$ , ${(A R)}^{3} = 1$ . The structure of the group however is given, at a glance, by the equations in the text.

³⁴Here again the group can clearly be deﬁned in terms of $A$ , $Q$ and $R$ .

³⁵On groups whose order is of the form $p^{2} q$ the reader may consult; Hölder, “Die Gruppen der Ordnungen $p^{3}$ , $p q^{2}$ , $p q r$ , $p^{4}$ ,” Math. Ann. XLIII (1893), in particular pp. 335–360; and Cole and Glover, “On groups whose orders are products of three prime factors,” Amer. Journal, XV (1893), pp. 202–214.

³⁶When it is necessary to call attention directly to the fact that the group we are dealing with is supposed to be presented as a group of substitutions, the group will be spoken of as a substitution-group.

³⁷The symmetric group has been so called because the only functions of the $n$ symbols which are unaltered by all the substitutions of the group are the symmetric functions. All the substitutions of the alternating group leave the square root of the discriminant unaltered (§ 11).

³⁸Jordan, “Récherches sur les substitutions,” Liouville’s Journal, 2^me sér. Vol. XVII (1872), p. 355.

³⁹Jordan, Traité des Substitutions (1870), p. 60.

⁴⁰“Récherches sur les substitutions,” Liouville’s Journal, 2^me sér., Vol. XVII (1872), pp. 357–363.

⁴¹Traité des Substitutions (1870), pp. 31, 32.

⁴²For a further discussion of the limits of transitivity of a substitution-group, compare Jordan, Traité des Substitutions, pp. 76–87; and Bochert, Math. Ann., XXIX, (1886) pp. 27–49; XXXIII, (1888) pp. 573–583.

⁴³On the subject of this and the following paragraph, the reader should consult the memoirs by Mathieu, Liouville’s Journal, 2^me Sér., t. V (1860), pp. 9–42; ib. t. VI (1861), pp. 241–323; where the groups here considered were ﬁrst shewn to exist.

⁴⁴The author has shewn (Messenger of Mathematics, Vol. XXV (1896), pp. 147–153) that the type of group considered in the text is the only type of doubly transitive group of degree $p^{m}$ and order $p^{m} (p^{m} - 1)$ when $m = 3$ ; and that, when $m = 2$ and $p > 3$ , the same is true. When $m = 2$ and $p = 3$ , there is one other type.

⁴⁵On intransitive groups, reference may be made to Bolza, “On the construction of intransitive groups,” Amer. Journal, Vol. XI, (1889), pp. 195–214. The general isomorphism which underlies the construction of these groups is considered by Klein and Fricke, Vorlesungen über die Theorie der elliptischen Modulfunctionen, Vol. I, (1890), pp. 402–406.

⁴⁶“Ueber die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodul,” Crelle, t. CI, (1887), p. 288.

⁴⁷Dyck, “Gruppentheoretische Studien, II,” Math. Ann. XXII, (1883), pp. 86–95.

⁴⁸Jordan, Traité des Substitutions, p. 65; where, however, the exceptional case is overlooked.

⁴⁹This result, stated in a somewhat diﬀerent form, is given, among many others, in the letter written by Galois to his friend Chevalier on the evening of May 29th, 1832, the day before the duel in which he was killed. The letter was ﬁrst printed in the Revue Encyclopédique (1832), p. 568; it was reprinted in the collection of Galois’s mathematical writings in Liouville’s Journal, t. XI (1846), pp. 381–444

⁵⁰The commas in the symbols for the substitutions are here used to prevent confusion among the one-digit and two-digit numbers.

⁵¹Jordan, Comptes Rendus, t. LXXIII (1871), pp. 853–857; ib. t. LXXV (1872), pp. 1754–1757. And see the Bulletin of the New York Mathematical Society, Cole, 1st series, Vol. II, pp. 184–190; 250–258: Miller, 1st series, Vol. III, pp. 168, 169; 242–245: 2nd series, Vol. I, pp. 67–72; 255–258: Vol. II, pp. 138–145. Quarterly Journal of Mathematics, Cole, Vol. XXVI, pp. 372–388: Vol. XXVII, pp. 35–50: Miller, Vol. XXVII, pp. 99–118; Vol. XXVIII, pp. 193–231. These memoirs were all published between 1892 and 1896; in them further references will be found.

⁵²The results contained in §§ 139, 140 are due to Jordan (Liouville’s Journal, Vol. XVI, 1871) and Netto (Crelle’s Journal, Vol. CIII, 1889). They have been extended by Marggraﬀ: “Ueber primitiven Gruppen mit transitiven Untergruppen geringeren Grades,” (Inaugural Dissertation, Giessen, 1892). With the notation used in the text, Marggraﬀ shews that, unless the symbols aﬀected by $H$ can be divided into imprimitive systems of $r$ symbols each, in at least $r + 1$ distinct ways, $G$ will be $(n - m + 1)$ -ply transitive. In particular, if $H$ is a cyclical group of degree $m$ , $G$ is $(n - m + 1)$ -ply transitive. He also shews that in any case $m \geq \frac{1}{2} n$ .

⁵³It is shewn in § 112 that ${P}$ is permutable with a circular substitution of $p - 1$ symbols, which leaves one symbol $a_{1}$ unchanged. If there are other substitutions which leave $a_{1}$ unchanged and are permutable with ${P}$ , some such substitution will leave two symbols unchanged. This is clearly impossible. Hence the group of order $p (p - 1)$ is the greatest group of the $p$ symbols in which ${P}$ is self-conjugate.

⁵⁴Comptes Rendus, t. LXXV (1872), p. 1757.

⁵⁵Netto: “Substitutionentheorie,” pp. 220–235; “Zur Theorie der Tripelsysteme,” Math. Ann. Vol. XLII, (1892), pp. 143–152. Moore: “Concerning triple systems,” Math. Ann. Vol. XLIII, (1893), pp. 271–285. Heﬀter: “Ueber Tripelaysteme,” Math. Ann., Vol. XLIX, (1897), pp. 101–112.

⁵⁶Klein, “Vorlesungen über das Ikosaeder und die Auﬂösung der Gleichungen vom fünften Grade” (1884), p. 232. Hölder, Math. Ann., Vol. XLIII (1893), p. 314.

⁵⁷Hölder, “Bildung zusammengesetzter Gruppen,” Math. Ann., Vol. XLVI (1895), p. 326.

⁵⁸Frobenius, “Ueber endliche Gruppen,” Berliner Sitzungsberichte, 1895, p. 183.

⁵⁹Frobenius, “Ueber auﬂösbare Gruppen, II,” Berliner Sitzungsberichte, 1895, p. 1027.

⁶⁰Frobenius, l.c., pp. 1028, 1029.

⁶¹Hölder, “Bildung zusammengesetzter Gruppen,” Math. Ann., Vol. XLVI (1895), p. 325.

⁶²Ibid. p. 325.

⁶³Hölder (loc. cit. p. 331) gives a theorem which is similar but not quite equivalent to Theorem VI.

⁶⁴Hölder, Math. Ann. Vol. XLVI, (1895), p. 345.

⁶⁵This and all subsequent congruences in the present section are to be taken, $(mod p)$ .

⁶⁶Hölder, loc. cit. p. 343.

⁶⁷“Ueber auﬂösbare Gruppen, II,” Berliner Sitzungsberichte, 1895, p. 1028.

⁶⁸Frobenius, loc. cit. p. 1030.

⁶⁹Frobenius, loc. cit. p. 1030.

⁷⁰The investigations of this Chapter are due to Dyck, “Gruppentheoretische Studien,” Math. Ann., Vol. XX, (1882), pp. 1–44. We have followed Dyck’s memoir closely except in two respects. Firstly, we have used a rather more deﬁnite geometrical operation than that of the memoir; and secondly, we have not specially considered a regular and symmetric division of a closed surface, apart from a merely regular division.

⁷¹The reader who refers to Prof. Dyck’s memoir should notice that the deﬁnition of the operation $S_{r}$ above given is not exactly equivalent to that used by Prof. Dyck. With his notation, the white polygon here considered would correspond to the operation $S_{p} S_{q} \dots S_{r} S_{s}$ .

⁷²In ﬁg. 4 the orthogonal circle, which is not shewn, is taken to be a straight line.

⁷³Forsyth, Theory of Functions, p. 325.

⁷⁴Hurwitz, “Algebraische Gebilde mit eindeutigen Transformationen in sich,” Math. Ann. XLI, (1893), p. 426.

⁷⁵Hurwitz, loc. cit. p. 424.

⁷⁶Forsyth, Theory of Functions, p. 330.

⁷⁷Klein, “Vorlesungen über das Ikosaeder,” Chap. I.

⁷⁸Dyck, “Gruppentheoretische Studien,” Math. Ann. XX, (1882), p. 35, and Klein, loc. cit. p. 26.

⁷⁹Dyck, “Ueber Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemann’scher Flächen,” Math. Ann. XVII, (1880), pp. 501–509.

⁸⁰This agrees with the result as stated by Dyck, “Gruppentheoretische Studien,” Math. Ann. Vol. XX, (1882), p. 41.

⁸¹American Journal of Mathematics, Vol. I, (1878), pp. 174–176, Vol. XI, (1889), pp. 139–157; Proceedings of the London Mathematical Society, Vol. IX, (1878), pp. 126–133.

⁸²For further illustrations, the reader may refer to Young, Amer. Journal, Vol. XV, (1893), pp. 164–167; Maschke, Amer. Journal, Vol. XVIII, (1896), pp. 156–188.

⁸³The homogeneous linear group and its sub-groups forms the subject of the greater part of Jordan’s Traité des Substitutions. The investigation of its composition-series, given in the text, is due to Jordan. The complete analysis of the fractional linear group, deﬁned by

y \equiv \frac{α x + β}{γ x + δ}, (mod p),

where

α δ - β γ \equiv 1,

is due originally to Gierster, “Die Untergruppen der Galois’schen Gruppe der Modulargleichungen für den Fall eines primzahligen Transformationsgrades,” Math. Ann. Vol. XVIII, (1881), pp. 319–365. With a few unimportant modiﬁcations, the investigation in the text follows the lines of Gierster’s memoir. A similar analysis of the simple groups of order

2^{n} (2^{2 n} - 1)

, which can be expressed as triply transitive groups of degree

2^{n} + 1

, has been given by the author, “On a class of groups deﬁned by congruences,” Proc. L. M. S. Vol. XXV, (1894), pp. 132–136.

⁸⁴The operation $(z x_{1}, x_{2}, \dots, x_{n})$ has $z$ for its determinant.

⁸⁵These and all succeeding congruences are to be taken $(mod p)$ , unless the contrary is stated.

⁸⁶For the case $n = 3$ the reader may consult a paper by the author “On a class of groups deﬁned by congruences,” Proc. L. M. S. Vol. XXVI, (1895), pp. 58–106.

⁸⁷A non-cyclical group of order $4$ is called a quadratic group.

⁸⁸This is another of the results stated in the letter of Galois referred to in the footnote on p. 1106.

⁸⁹Moore: “On a doubly-inﬁnite series of simple groups,” Chicago Congress Mathematical Papers, (1893); Burnside: “On a class of groups deﬁned by congruences,” Proc. L. M. S. Vol. XXV, (1894), pp. 113–139.

⁹⁰Hölder, Math. Ann. Vol. XL, (1892), p. 57.

⁹¹Frobenius, Berliner Sitzungsberichte 1895, p. 190, and Burnside, Proc. L. M. S. Vol. XXVI, (1895), p. 209; for the case where $α ≯ m$ .

⁹²If this sub-group is cyclical, then $Q = {Q'}^{p}$ .

⁹³Proc. L. M. S. Vol. XXVI, (1895), p. 199.

⁹⁴“Die Gruppen mit quadratfreier Ordnungszahl” Göttingen Nachrichten, 1895: pp. 211–229. Compare also Frobenius, Berliner Sitzungsberichte, 1895, pp. 1043, 1044.

⁹⁵“Über auﬂösbare Gruppen, II”: Berliner Sitzungsberichte, 1895, p. 1035.

⁹⁶Frobenius, loc. cit., p. 1041.

⁹⁷Frobenius, “Über auﬂösbare Gruppen, II” Berliner Sitzungsberichte, 1895, p. 1039.

⁹⁸Burnside, Proc. L. M. S. Vol. XXVI, (1895), p. 332.

⁹⁹The reader will easily verify that, when this condition is satisﬁed, $\frac{H}{h}$ is Abelian, $h$ being the sub-group formed by the self-conjugate operations of $H$ .

¹⁰⁰Math. Ann. Vol. XL, (1892), pp. 55–88.

¹⁰¹American Journal of Mathematics, Vol. XV, (1893), pp. 303–315.

¹⁰²Proc. London Math. Soc., Vol. XXVI, (1895), pp. 333–338.

¹⁰³It is not, of course, here suggested that $2835$ is the order of a simple group. It may, in fact, be shewn with little diﬃculty that no number of the form $3^{4} p q$ , $3^{3} p^{2} q$ , or $3^{2} p^{3} q$ , where $p$ and $q$ are odd primes, can be the order of a simple group. A much higher limit than $2835$ may therefore be given for the order, if odd, of a simple group.

¹⁰⁴“Bildung zusammengesetzter Gruppen,” Math. Ann. Vol. XLVI., (1895), pp. 321–422; in particular, p. 420.

¹⁰⁵This term is used by other German writers to denote the holomorph of a group of prime order.

¹⁰⁶It is to be noticed that, if $G$ is multiply isomorphic with $H$ , then $H$ is “meriédriquement isomorphe” with $G$ .

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