CONCERNING JORDAN'S LINEAR GROUPS.

Presented to the American Mathematical Society, August 28, 1895.

BY ELIAKIM HASTINGS MOORE.

Introduction.

=

I PRESENT to the AMERICAN MATHEMATICAL SOCIETY to-day a continuation of the paper* presented last November, entitled The group of holoedric transformation into itself of a given group. To recall briefly: The given (abstract) group G of order n has the elements s1 identity, 82,.. Sn The substitution-group" of transformation of G into itself is the substitution-group on the n letters s1,.. s, which leaves invariant the multiplication-table for G. Letters s which are conjugate with one another under " must as elements of G1 have the same period. Thus, s, identity is invariant, and I" is really - on the n-1 letters 82,.. Sn

=

We are to consider to-day the case that F-1 is transitive on the n - 1 letters 82,.. S. Then the n-1 elements 82, . . S of G have the same period, which must then be a prime p. Hence G has the order n = p". Every group G-p" has, in accordance with an important (Sylow's) theorem,† at least one element different from identity commutative with every element of the group. This property of the element may be read out of the multiplication-table for G, and is hence invariant under . But I'"-1 is transitive on the n 1 letters 82,.. Sn Hence every element of G-p" is commutative with every other element. Our given group G is then the Abelian G, or rather, omitting the', G with n generating elements, each of order p, and commutative with one another. It will cause no confusion if we refer to it hereafter simply as the Abelian GTM”.

*Bulletin of the American Mathematical Society, ser. 2, vol. 1, pp. 61-66, Dec. 1894.

Mr. HÖLDER explained this notion of the group of holoedric transformations into itself of a given group, for use in his memoir: Die Gruppen der Ordnungen på, pq2, pqr, p1 (Mathematische Annalen, vol. 43, pp. 301412; see pp. 313, 314), which bears the date March 28, 1893. We, however, hit on the notion independently of each other; see the foot-note (**) of p. 66 of my former paper.

SYLOW: Mathematische Annalen, vol. 5, p. 588.

Гри

Ω(p)

§ 1.

of holoedric transformation into itself of the

n

The group Abelian group G is Jordan's linear homogeneous substitutiongroup of degree p", LHG)"

(1)

For the Abelian G," we take the n generators

with the complete system of generating relations.

(2)

ap = 1,

(i = 1, 2, .. n)

(i, j = 1, 2, . . n)

=

where the suffixes and exponents k are integers taken modulo p, and where K is a symbol standing for (k1, k2, . . kn). The general multiplication equation is

It turns out that the general substitution og of by SS'1, 2, · · X′n' where

replaces 8x=821, 22, · · In

whose elements 9, are integers taken modulo p. [To follow the customary notation we should write congruences (modulo p) everywhere instead of equations. But in group-theoretic applications such as the present, it is much better to breathe the spirit of the congruence once for all into the definitions of the symbols and operations.] Hence, indeed, I", is Jordan's linear homogeneous substitution-group* of degree p", LHG ph of order †

(8)

N(pr)

Q (p") = (q′′ − 1) (q′′ − q) (q′′ — q3) · · (q” — q′′−1).

* JORDAN: Traité des substitutions, p. 92, 1870.

† JORDAN: loc. cit., p. 97.

D(pn),

Q(pn)

N(pn)

This identification of the r of the Abelian G," with the LHG I obtain first by holding the G, as an abstract group; I omit the details of this identification. We may however take the G, concretely as the regular Abelian substitutiongroup G on the p" letters s = 8x1, x2, .. ; the general element (3) $x = Sk1, kq, kn then (4, 5) replaces sy by sr, where

(9)

ph

pn

X' = X +K, x';= x;+k; (i = 1, 2, . . n).

We thus win direct contact with Mr. Jordan's work. The G (9) is within the symmetric substitution-group on the p letters & self-conjugate under the linear non-homogeneous group LGP of degree p" and of order p′′N(p"), whose general substitution σG, & replaces sy by sy, where

pnQ (pn)

o K

j=n

(10) X'=GX+K, x'=Zqwpx;+k; (\I1\‡0) (i, j=1,2,..n).

σG. K replaces §Ã1, Sk1, §Ã ̧ by §Ã ́ ̧, SK2, SK ́ý where

K'1 = GK1+K, K'2 = GK2+K, K's = GK, + K,

so that

K'1+ K'2- K's = G (K1 + K2 — K3) + K;
K'1⁄2 — K' ̧

hence under σG, K

of the LG pan(pa) (10) a multiplication equation of the G, 8, 8, 8K3=8K1+K2 (4, 5) is preserved, that is,

=

SK SK2SK3= SK'1+K ́2

pnQ(pn)

if and only if K=(0)=(k1, k2, . . k„) = (0, 0, ..0), that is, if and only if the substitution oG, K of the LG (10) is a substitution "G, 0G of the LHG (6). We have then this (second) identification of the p of the Abelian G," with

the LHG

The

Q(pn)*

Q(pn)

group Iph letters 8(X+(0)).

p" - 1 2n 1 letters. imprimitive; the letter S

LHG (6) is transitive on the p"-1 For p 2 it is doubly transitive on the For p > 2 it is simply transitive and 821, 22..z, belongs to and by the 8x=821, In ratios of its n suffixes X=(x1: :: a) determines the system of imprimitivity containing the q-1 letters * s(l=1, 2,..p-1); in the G the elements sx and the identity Sox=8(0) constitute the cyclic group G,{s} determined by s say the G, Thus,

* X = (X1, ・ ・ Xn), 1X =(l£1, ・ ・ lXn).

the I

Q(pn)

Q(ph)

=LHG permutes first the (p" — 1)/(p − 1) G, of G, and afterwards fixes the elements within the various groups. The self-conjugate sub-group which keeps every G, fixed is of order p—1:

(11)

{X'=lX, x';= lx; (i = 1, 2, . • n)},

(= 1, 2,..p-1).

The quotient-group, which is a substitution-group on the (p-1)/(p-1) G,, has the order (p")/(p-1). Analyti cally, it is the LHG) taken fractionally; that is, the linear fractional group† LFG-1)/(-1), whose general substitution OG replaces the Gp,x by the G., where

σα

(p)/(p-1)"

LCf[p], LHCf[p" -1], LFCƒ[(p" − 1)/(p − 1)]: connected with the Abelian Gn are defining invariants respectively for the three linear groups:

The notion configuration I transfer to tactic from geometry † ; for the proof and ultimate statement of the theorems about to be stated with utmost brevity, this notion must be used to its full content; to-day, however, the term tactical configuration shall be merely a name.

The linear configuration LCf[p"] in p" letters.

The p" letters of the LCf [p"] are the p" elements s of the Abelian G". The G contains (p" -1)/(p-1) sub-groups,

JORDAN: loc. cit., p. 228. In my notation the two subscript dots (..) are the ratio dots (:), and are to call to mind that we may without changing anything replace X (X1, • Xn), X' = (x'1, ・ ・ X'n), G = (94) by 1X = (1x1, lxn), l'X' (l'x'1, ・ ・ V'x'n), mG = (mg), respectively, where 1, l', m are any integers taken modulo p, but 10, l' ‡0, m‡0. † See, for instance, REYE: Das Problem der Configurationen (Acta Mathematica, vol. 1, pp. 92–96, 1882).

G-1. With respect to each sub-group G-1 the p" elements s of the G are exhibited as a certain rectangular array of p lines with p-1 elements in each line; the order of the lines and the order of the elements in each line are immaterial; one line contains the p-1 elements of the G-1 itself. We separate every array into its constituent lines, and have before us in the system of (unordered) p(p” — 1)/(p − 1) lines or combinations of p-1 letters each the linear configuration in pr letters, LCf[p"].

This LC [p] for n ≥2 defines, as the maximum substitution-group on the p" letters sy leaving it invariant, exactly the LG) ($1 (10)).

The linear homogeneous configuration LHCf[p" -1]

in pr - 1 letters.

The p" — 1 letters of the LHCf [p" - 1] are the p” — 1 elements s(X(0)) of the Abelian G", the identity so excepted. The LHCf[p"-1] is obtained from the LCf[p"] by omitting every line or combination containing the discarded letter s(0). The LHCf[p" - 1] consists, then, of a system of p" - 1 lines or combinations of p-1 letters each. This LHCf[p” — 1] is tactically self-reciprocal, that is, we can distribute a notation s' to the p-1 lines in such a way that the LHCf[p” −1] on the p"-1 letters 8 as grouped by the p" — 1 lines s'x differs only in the priming () from the LHCf[p" - 1] on the p"-1 lines s'y as grouped by the p"-1 letters s

Q(pn)

*

This LHCf[p" - 1] for n ≥ 2 serves as a defining invariant for exactly the LHG or pn-1 (§ 1, (6)). The self-reciprocity of the LHCf [p" - 1] establishes an holoedric isomorphism of the LHG with itself. This isomorphism is (at least for n≥3) not* that arising from a transformation of the LHG through one of its own elements.

1

N(pn)

-1

Q(ph)

The linear fractional configuration LFCf [(p"-1)/(p-1)] on (p" — 1)/(p − 1) letters.

The (p"-1)/(p-1) letters of the LFCf [(p" - 1)/(p − 1)] are the (p"-1)/(p − 1) cyclic groups G; x of the Abelian G The LFCf[(p-1)/(p-1)] is obtained from the LCf [p"] by

p'

Notice the particular case (q = 2, n = 3) in § 2 of my paper cited above. The LHCƒ [28 − 1 = 7] and the A7 are, so to say, complementary. Indeed, for q = 2, n = any, the LHC [2"-1] determines uniquely a A2"-1, from which the LHCf [2′′ – 1] is likewise uniquely determined. This A2"-1 serves as a defining invariant for the LHG2"-1

Ω (2)