on the primitive substitution groups of degree sixteen

On the Primitive Substitution Groups of Degree SixteenAuthor(s): G. A. MillerSource: American Journal of Mathematics, Vol. 20, No. 3 (Jul., 1898), pp. 229-241Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369822 .

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On the Primittive Substitution Groups of Degree Sixteen.

BY G. A. MILLER.

The object of this paper is to determinie all the primitive groups of degree 16 and to study them in regard to solvability. It will be seen that the number of these groups is considerably larger than it hlas been supposed to be.* The paper is divided into two sections. In the first we deteriiine the grotups by very simple methods and study them with respect to the given property. In the second we prove that it is impossible to constrtuct a primitive grolup of degree 16 that is not giveni in the first section.

?1.

Detergmination of thte groztJs.

It will be proved that all these groups, with the exception of tlle two that include the alterniating grotip of this degree, conitaini a self-conjulgate subgroup of order 16. As a self-conijugate suibgroup of a primiiitive group is transitive, this mitust be regular. Since the entire grouip must transformn each of its stubstitutioils except identity into a system that genierates it, it mulst be the Abelian grolup that contains 15 subgroups of order 2. We slhall denote it by H and suppose that its substitutions are

1 ai . bj. ck . dl. emn. f.o1 goI I, ab. cd. ef. gh. l. kl. mn . op B, (Qj. bi . cl . c7k . ena . fqn .g p .o lo J, ac. bd. eg.fh. ik-.jl. mo. np C, ak . bl. ci. c7j. eo.fp).gn? .7ln K,

ad . bc . eh . fg .i il j7 . mjk . m no D, al. bk7. cj. di. ep. fo. ga. 7am L, ae. bf. cg. dh. im.jn. ko. lp E, ar. bn. co. dlp . ei.fj. gk7.7l h lJi, cf. be . ch . clg . din . jm . 7]) . lo F, an . bm . cp . do . ej. fi. gl .1k N, ag.bh .ce.clf.io.jlj .i.km . In G, ao.bp).cm7z.c1n.ek.fl.gi.1aj 0, ah. bg. cf. cle. ipi.jo . kn. lm H, ca) . bo . cn. dm . el. fk . gj. hti P .

* Only 12 of these groups that do not contain the alternating group are given in Jordan's enumeration, Comptes rendus, vol. 75, p. 1757. We shall prove the existence of 20 such groups.



230 MILLER: On the Primitiv?e Substitution Groulps of Degree Sixteen.

The group of isomorphismns of H* is evidently doubly transitive, of degree 15 and order 15. 14. 12. 8 = 8 ! * 2. It is known that there is only one group

that satisfies these coniditions, and that it is simiiply isomorphiic to the alternating group of degree 8 (K). To every subgrouip of KI corresponds a transitive group of degree 16 that conitains H as self-conjugate subgrouLp. Its subgroup that includes all its substitutioIns whichl do not inivolve a given elemnent is simliply isomorphic to the corresponding subg,roup of K. We shall confinie our attention to the primitive groups.

The subgroup of the group of isomorphism that corresponds to a primitive group must tranisform each of the substitutions of H, excepting identity, into substitutions that generate H. It mnust tllerefore be of degree 15, and its order

canniot be less tlhani 5. To a subgroup of order 5 in IK corresponids a group of

order 80 thlat contains H as self-conijugate subgroup. This group ( G1) miust be primitive, for it contains no self-conjugate subgroulp besides H and 1, and there is no tranisitive group of degree 8 and order 80. It is solvable since its factors of com1position are 2, 2, 2, 2, 5. It may be generated by H anid

boeje . dpkhnl .fhgim .t

From the preceding paragraph it follows that to every subgroup of K whose

order is divisible by 5 there corresponds a primitive grotup of degree 16 that conitains H[ as self-conijtugate subgroup. If we incluide K7, there are 16 such suib-

groulps, viz. (abcde) cyc.,J (abcde)10, (abcde) cyc. (fght) cyc., {(abede)20 (fy) pos.,

(abcdle)l0 (fqh) cyc., (abede) pos., (abcdef)61, (ac de)90 (fghl) all pos., {(abede) all

(fg) lpos., - (abedef/)1,0 (ght) } pos., (abede) pos. (Ayh) cyc., (abedef ) pOS., I (abede) all

(fgh) all} pos., I (abedef) all (gi t pos., (abcdefg) pos., (abcdefgh) pos. As all the

subgroups of K that are similar to any one of these are conjugate, there cani be nio more than 16 primitive groups of degree 16 that conitain H as self-conijugate

subgroup anid whose orders are divisible by 5. We proceed to prove that no two of these 16 primitive groups are simply isonmorphic and to find their factors of comiipositioni and generating substitutions.

* H6lder, Mathematische Annalen, vol. 43, p. 314. t The substitutions that are to be added to H to generate the required groups permuute the substitu-

tions of H in exactly the same manner as their own elem-ents; i. e. they have been selected in such a wav they they are the sanme as the corresponding ones in capital letters.

t Cayley, Quarterly Journal of Matlhemllatics, vol. 25, p. 71.



MILLER: 0Oz the Pr-imitive Substitution G-roupcs of Degree Sixteen. 231

The grotup of order 160 (G2) may be generated by G1 and

bj.dl.eo.fhk.gm.nqp.

Since it contains G6 as maximal self-conjtugate subgroup, its factors of composition nmay be found by adding 2 to those of GI. Hence it is solvable. The group of order 240 (G3) miiay be generated bv G1 and

bmn . cik . deh, . flo . gp-j.

As it containis G1 as maximal self-con-jtugate subgroup, its factors of composition may be obtainied by addinig 3 to those of G1. It evidenitly containis 16 conljuogate subgrotups of order 15. The groups of orders 320 and 480 (G4, G5) may b)e gener-ated, respectively, by G2 and

bejo .dglm .fqzh .ik, blmn . cik. deh .flo .g)j.

Sinice G2 is a imaximal stubgrotup of each of these groups, their factors of comnposition may be obtained by adding 2, 3 respectively to those of G2. Hence they are solvable. G5 contains G3 as self-conjugate subgroup.

Since Kcontain-s 3 subgroups of order 60, there are 3 primitive grouLps of degree 16 and order 960 that contain H as self-conjugate subgroup. Only one of these (G6) is solvable. It may be generated by G5 and

bejo . cglgm . fn lap . ik.

It is solvable since its factors of comlposition nmay be obtained by addinig 2 to those of G5. We have now found the 6 solvable priiitive grotups of degree 16 whose orders ale divisible by 5. The remiaining 12 groups wlhose orders satisfy this condition are insolvable.

The remainilng two groups of order 960 correspond to

(abdedf)60, (abccle) pos.

in K. Hence each of tlheimi conitainis only two self-conjugate subgroups, viz. Haind 1. Their factors of comi)positioni are 2, 2, 2, 2, 60. The former (G7) may be generated by G2 and

blpo . cnud . ekl. gli .

The latter (G8) 1nay be generated by H anid

7of ___7.. _pg_ . 7'~jk bo7p. ey. i7 . ]? a _h7 7.



232 MILLER: Oni the Primitive S-ubstittution Groutps of Degree Sixteen.

Since the substitutions of order 3 in G7 permnute only 12 of the substitutions of

H while those of G8 permute 15, these two groups are not simply isomnorphic.

The two prinmitive groUps of order 1920 (G9, G10) correspond to

(abcdef)120 (g7a) I pos., { (abede) all (fg) I pos.

in K. Hence eaclh of them conltains 3 self-conjugate subgroups, viz. H, 1 and

the one of order 960. Their factors of compositioni are obtained by adding 2 to

those of the preceding two groups. They may be generated by addinlg

bc1jl. ck . epon . fnihg, bpch . di. ejmk .fgon

to G7 and G8 respectively. They canniot be simiply isomorplhic since their self-

conijugate subtgroups of order 960 do not have this property.

The group of order 2880 (G1,) may be generated by G8 and

brnu. dii. ceg. hoj.fkhp.

Its factor-s of comiiposition may evidently be obtainied by adding 3 to those of G8.

Tlhe two groups of order 5760 (G12, G13) correspond to

I(abcde) all (fghl) allI pos., (abedef) pos.

Since the former contains GU1 as self-conjugate subgroup, its factors of composi-

tion inay be obtained by adding 2 to those of Gll. This is the last of the six

primitive groups of degree 16 whose solution depends only upon that of the

alternating group of degree 5. It may be generated by G1I and

blpch . di . ejmnk .fJouz .

The factors of compositioni of G13 are evidently 2, 2, 2, 2, 360. It may be

generated by G7 and bop-1 . cge . dil . fmj. hMe.

For

bnitloj. cp)e7kd .fhimg X bj . dl . eo .flh. gm . qp = bpo . czd . eki . ghi, bp.,o . ceg . dli .4f/n . hun7 X bp1o . cutd . ekil. gh i = bop. cki . delh .fjrm . gni .

Hence the given genierator corresponds to a substitution in the same elements in K as G,. It clear ly corresponds to a stubstitution of degree 3.

The group of order 11520 (G14) conitains GU as selr:conjugate subgroup. Its

factors of comiiposition inay therefore be obtained by adding 2 to those of G,3. It

is genierated by G13 and bdjl. ck . epon . fmlhg.



MILLER: On the Primitive Substitution Groutp)s of Degree Sixteen. 233

G13 and G14 are the only two primitive groups of degree 16 whlose solutioni depends only upon that of the alterniatin-g grotup of degree 6 but not tupon any one of its subgroups.

The group (GU5) wlhiclh corresponids to the alterniating group of degree 7 in K inay be generated by H and

bioj4jg . clnmeZf, bcd . cni .fol. gr lin.7z.

For the former of these generators is transformiied inlto its square by

el-i . fkmi . gjp .lhio.

Since the latter is comnmutative to this, it tmutist corresponid to a substitution of degree 3 in K hliose elemienits are included in the substitutionl of degree 7 to wlicih the formner generator corresponds. The factors of conmpositioni of G15 are 2, 2, 2, 2, 2520.

The lLargest group that contains H as self-conjuglate subgroup (G16) inay be generated by Gln and

bd . fl . jl. np .

For the alternating group of degree 8 contains only two types of substitutions of order 2. We hlave seen that the substitutions whiclh correspond to tllose of the type ab . cl perinute 12 substitutions of If. The given generator nmustt therefore corresponid to a substitutioni of degree 8 in K. The factors of conilposition of G16 are 2, 2, 2, 2, 20160.

We have niow conisidered all the possible pr-imitive groups of degree 16 that con-tain Has a self-conijugate subgroup anid whose order is divisible by 5. We have seen that 6 of these 16 groups are solvable, 6 others depend Lupon the so]u- tion of the alter nating group of degree 6, wlhile the reiimaining fouir depend uipon the solutioni of the alternating groups of degrees 6, 7 and 8. There alre 4 addi-

tional primnitive groups that confiftin H as a self-conjugate subgrotup. We proceed to consider these.

Since the groups

(ae. If ce. dhl)(abc) cyc. (efg) cyc.,* (ace . f . cg . dlh) J (acb) all (efg) all pos.,

(afbe . cg. (h) (abc) all (efg) all pos., (ace. bf cg . dh) (abc) all (efg) all

are mnaxinal suLibgroups that do nlot conitain any self-conjuigate stubgroup besides

*Cayley, loc. cit.

31



234 MILLER: On the Primitive Substitution Group)s of Degree Sixteen.

identity of the four groups, in order,

(ae. bf. cg.d h)(abed) pos. (efgli) pos., (ce . bf . cy. dh) (abed) all (efgla) all pos., (afbe. cg . dh) I (abecl) all (efyh) all pos., (ae. 7f. g . dh)(a bed) all (efgh,) all,

each of the latter four groups is simply isomorphic to a primitive group of degree 16.* As each of these grouips is evidently solvable, its factors of comnposition are the samie as the primne factors of its order.

The first (GUn) mav be generated by H and

bjofnm . oeg. cdnihlk, bo. cnd . ek1i . gh.

Henice all its self-conijugate subgroups, except identity, conltain H. From this it follows independently tlhat it is primitive. G18 may be generated by Gl, and

bmnp1jof. ekl . dgh-len. It con tainis G,, as self-conjugate subgroutp. It is of orde r 576 while G,, is of order 288. G19 is of the samie order as G18. It mnay be generated by Hanid

bTo . c?1d .ld .gh,i, bnz . cide .fojp. gklh.

G20 contains G 1, G, l as self-conjugate subgroups. It is of order 1152 and may be genierated by G, anid

bjopfmn . ceg. dnilalk.

We lhave now found the 20 priMitivNe groups of degree 16 that do nlot contain the alternating group of this degree, and have seen that 10 of theni- are solvable while the remiainiing 10 are insolvable. Since the given genierating substitutions, excludinig f, do nlot conitain a, they generate a subgroup GI whlose order is obtainied by dividing tlhe order of the group by 16. If GI is a timies transitive, the correspondinggr otip is a + 1 times transitive. The class of G' is the samie as that of the corresponidinig group, etc. The G' of GI, is the group of isomoiphisms of H. By addinag the alternatinig anid the symnmetric group to the preceding we obtain the 22 primnitive groups of degree 16. The last two are evidently unsolvable. Their factors of composition are respectively 16! + 2; 2, 16! 2.

? 2.

Proof that there are no other primn-itive groups of degree sixteen.

As we have examined all the possible groups that conitain H as a self-conjugate subgroup an-d wlhose orders are divisible by 5, we do not need to consider,

*Dyck, Mathematische Annalen, vol. 22, p. 102.



MILLER: On the Primitive Substituttion Gro?up)8 of Degree Sixteen. 235

in what follows, the groups whiclh satisfy these two conditions. We shall consider the siinply transitive and the multiply transitive groups separately, begin- ining with the foriner. The group in question will generally be represented by G and its subgroup which conitains all its substituLtions that do not contain a given element by G'.

A.-Si?rnp_7y transitive groupus.

We slhall inake frequent use of the followinig theorems: Theorem I. GI cannot contain a transitive sub group. Theoremr II. All thze pr imee 'fl.umbe'r 'wd/ (al. -ivid7e thze ordler of one of thze tran8i-

tive conlstituents of G', divide the orcder of every oilher constitutient. Thleoretn III. If a transitive constititent of GI is of a prime degree, all its oth7er

transitive constidtents acre of ant equal or a larger clegree. Theorem IV. If pja is the highest po?ower of ac prime number thcat is contaccined

in the order of G, the s8ubgroiipjs of order pa are tranjsfornzed by the substitutiowns of G according to a transitive group whose order is divisible by pt butt not by 1 +1. G conltainls a self-conju.gate sebgyrouzpl of order pa-f3

Tlteo rem V. If G' contains a set/conjugate subgroup (H) of degree -a, n being the degree of G, H' mnu.st be t7e transformn, with resp)ect to stubstituttions of G, qf a - 1 other s8ubg-roujp8 of GI (Hi, H2, . .. . , Ha'L). The 8UbstdUtiofls of G that transformn H', (g = 1, 2, ..., a- 1), inzto 1' transform also all the sutbstituttions

of G' that ar-e con in? utative to H. into st8bstittUtions of G'. Theoremin VI. GI transforms Hll', f', ...., a/- 1 according to the elemnents in

one of its constitt.cents of degree a - 1, and no two of these subgro?utps can have all their elements tin common, nor can any of themn contain all the elements of H'.

Theorem VII. The gqroul- generated by H1I, H2', . , E L1 is of degryee n -1, and GI is a rnaximal s8bgroup of G.

Theorem VIII Every se1f-cowjtgate subg'roup of a primnitive group) is trcaq- sitive.

GI contains a transitive constitutent of degree 12.

The other constituent is the symmetric group of degree 3. To 1 in this group must correspond an intransitive subgroup of the constituent of degree 12 (H'). The systems of intransitivity of HI are systems of non-primitivity of the constituenit of degree 12. These systems could not be of degree 2 since they would have to be transformed according to a regular group, and H' would then contain all the substitutions of G0 whose degree < 13.



236 MILLER: On the Pr imitive Sit6bstituttion Groitps of Degree Sixteent.

The given systemiis could not be of deoree 3 since they would have to be transformed according to a tranisitive group whose order is divisible by 4. As the degree of the systemiis of H' could evidently nlot exceed that of a conistituenlt of G', it is impossible to conistruct a primitive group of degree 16 in whiclh '

containis a transitive coinstittient of degree 12. From one of the given tlleorems it follows directly that G' coulld not contain a tranisitive constituent of degree 1 1.

GI contatins 8a transitive consti?tuent of degree 10.

The constituent of degree 5 must clearly be transitive, anid its order must be the same as the order of G'; i. e. G' must be obtained by establishing a simple isomorphism between a transitive group of degree 10 and one of degree 5. When the order of G' is 5, 10, or 20, the subgroups of order 2a in G must be trains- formned by its stubstitutions according to a transitive group of degree 5. Hence there m11ust be a self-conijugate subgrouip of order 21, 3 > 0, in G, according to the giveni theoreimi. Sinice the substitutions of order 5 are of degree 15, we lave the congruence

20=1. mod. 5.

As c < 8, z3 _ 4. We have considered all suclh glroups.

When G' is of order 60 or 120, G must transforn its subgroups of order 2a

according to some transitive group of degree 5 or 15. In the former case G contains a self-conjugate subgroup of order 16, as we have just proved. In the latter case tlie corresponding group of degree 1 5 miiust be inon-prinmitive, since the orders of the primiitive groups are divisible by 9. Since such a non-primnitive group could conitaiin no substitution besides identity that leaves all its systemiis unchanged, it must be simply isomorphic to a transitive group of degree 5. Hence the preceding proof applies also to this case.

GI contahis a. transitive constitufient of degree 9.

Sinice the order of G' cannot be divisible by 5, the constitueint of degree 6 imiust be eitlher inon-primiiitive or intranisitive. If the order of G' would exceed that of the constituent of degree 6, the constituient of degree 9 would be non- primlitive anid GI would contain aii H' of order 3a. As the substitutions of order 3 and degree < 9 in the constituent of degree 9 are comnm-autative, those of order 3 in the constituent of degree 6 could not be commutative, i. e. this con- stittient wotuld be trainsitive alid it would contain 4 conjugate subgroups of



MILLER: Ont the Primitive Siabstitation Groups of Degree Sixteena. 237

order 3. As this is clearly imipossible, the order of G' is the same as that of the constituent of degree 6.

The order of G' cannot exceed that of the constituent of degree 9, since the order of the quotient groUp of the constituenit of degree 6 with respect to a suitable self-conijugate subgroup would not be divisible by 9. Hence G' must be composed of two simply isomorplhic grotups whose degrees are 9 and 6. The latter mlust be transitive, for if it were intransitive all its subgroups of order 3 and degree < 15 would be self-conjugate.

From the preceding it follows that G' miust be of order 18, 36, or 72. G inust therefore contain 8 conijutgate subgroups of order 3 anid degree 12. As none of its substitutions besides idenitity can transform each of these subgroups into itself, it miust be simnply isomorphic to a transitive group of degree 8 anid order 288, 576, or 1152. There are onily 4 such tranisitive groups. We have seen that each of them contains one miiaximal subgroup that does not include ally self-colnjtugate subgroup besides identity and whiose order is obtained by dividinlg the order of the group by 16. In other words, we have seen that each of these groups is simply isomorphic to one primitive group of degree 16. As each of the given groups of degree 8 contains only one set of conjugate subgroups of thle equired type, each of them is siiiply isomorphic to onily one prinmitive group of

degree 16.

GI contains a transitive constituent' of degree 8.

It is clear that the conistituent of degree 7 must be transitive. Since the two transitive conistituents of G' would be primitive, they would have to be silmiply isomorplhic. Hence G' would be of order 168. Its substitutions of order 2 would be of degree 12. The 6 systems of such a suibstitution are tralnsfornmed according to I (abcd)4(ef) } din. by the substitutions of G'. The substitutions of G would have to tranisformr these systemis according to a transitive group of degree 6 anid order 16. This is clearly impossible.

If G' would conitain a transitive constituent of degree 7 the other would hiave to be transitive and of degree 8. We have just seen that this is imiipossible. We have now considered all the cases wlhen U' contains a constituenit whose degree exceeds 6. The renmaininig cases do n1ot lead to any additional group, and are so simple that it seems unnecessary to consider them here.



238 MILLER: On the Primnitive Substitution Groups of Degree Sixteen.

B.-Muiltipiy transitive groups1.

We shall begin with the cases when G' is nion-primitive and contains an

intranisitive self-conjugate subgroup (H1). If H1 contains 3 systenms of intransi-

tivity it miust be comuposed of three simaply isomorphic transitive groups of degree

15, since its order cannot be divisible by 25. If its order is 5, 10, or 20 it mlust

contain a self-conijugate subgrouLp of or der 16, as has been proved above for a

similar case. Its order could not be 60 or 120, since the nulmber of its sub-

groups of order 3 and degree 9 would not be divisible by 7. If H1 conltains 5 systelm-s of initransitivity its subgroup of order 3a nmust

satisfy onie of the two conigruences

3a _ 1 mod. 5, 3 = 3 milod. 5,

sinice a substitution of order 5 in H1 could not trainsformi two of its subgroups of

older 3 inlto tlhemselves. Hence a = 0, 1, or 4. If a wele 4, H1 would conitain

5 or 10 subgroups of order 3 and degree 6, for onily 20 such subgroups are founid

in the grotup of order 35, and the two of the same degree could not occur in H1.

In G each of these subgroups would lhave to be transformed inlto itself by a

subgroup whose order is divisible by 5. This is clearly impossible. Hence

a= 0 or 1. We have seen that G contains a self-conjugate subgroup of order 16 when

a = 0 or when a = 1, and the systems of intransitivity of H1 are permuted by

G' according to the mnetacyclic group or one of its subgroups. Wlhen H1 is of order 3 and GI transformns its systems accordinig to the alternating group of

degree 5, its 3 substittutionis that colrespond to a substitution of order 3 in this

alternating group must be of order 3, two of them must be of degree 12 and the

tlhird of degree 15. All the substitutions of G' must therefore be commutative

with eaclh substitution of H1. Hence all its subgroups of order 12 that do not iniclude any self-conijugate subgroup are conjugate in two sets. Sinice one of

these two sim-ply isoimiorphic transitive groups* of degree 16 contains substitu-

tionls of order 3 and degree 9, we need to consider only one G'. We may suppose that it is generated by

adgjm)?belhkncfilo, adh . bei. cfg. mon.

The substitutionis of G that transform the latter of these generators inlto

itself miust permute its systems according to the alternating group of degree 4,

* Jordan, " Traite des Substitutions, " p. 272.



MILLER: On2 the Primitive Substitutito Group8s of Degree Sixteen. 239

since those of GI perm:ute theIm- according to the alternating group of degree 3. Hence and from the fact that bc. dlh . eg .if. k1. mn transforms the given G ilto itself, we miay suppose that G contains one of the followiig substitutions:

amdlohn bcefia ( i. andrnho bgecif ; 5jk.p aocdnhm bfegic

By trial we finid that only one of these 18 substitutions, viz. amdolhn . beefig .jp . kl, gener-ates a group whose substitutions that do not contaiii I) are conitaiined in G'. The cube of this last generating substitution and its transforms form the 15 substitutions differing from identity of a self-conjugate subgroup of order 16. Hence the group whiclh contains the G' that transformiis the systemls of H1 according to the syiimmetric group of degree 5 must also cointain a self-conjtugate subgoroup of order 16.

There cotuld be no primTiitive group containinig the H1 of order 6 since such a group would have to contain one of the two precedinig as self-conijugate subgroup. This is impossible, since the sutbstitutions of degree 12 and order 3 in G' could nlot transform the negative substitutions of H1 among themselves. Hence it renmains only to consider the cases wlhen G' is a primitive group of degree 15.

There are only 4 suclh primiitive groups that do not include the alternating grotup, viz. those whichl are simply isomorplhic to the symm-netric group of degree 6 anid the alternatiing groups of degrees 6, 7, 8. Eaclh of these groups containis substitutions of degree 12 and order 3. It will not be difficult to finid substitutions of G that are 11ot conitained in GI anid that are commlalutative to sluch a substitution. We slhall pursue this nmethod to find all tlle possible groups aild then prove that each of themii conitains a self-conjugate subgroup of order 16.

The GI of order 360 may be genierated by

abc. efgy. 7n. kkn, abef. bec7,g. imik'n .jl, adjkl. bmnei. eohfq.

Since the first of tlhese gen1erators is transformied into itself by 9 substitutions of G' which permute its first three systems cyclically, and all the substitutitions of this type are conjugate, G must conitain a subgroulp of order 36 that tranisfornms it into itself aind perinutes its systemiis ao.ccording to the alternlatilo,' group of degree 4. Hence it milust contain one of the followving substitutions:

aebfeg iiAylm7n (lii o pl- a/fbgee imk222 k (1o . 7h agbeef injk?nl I (Id-p. 1w



240 MILLER: Ont the Primitive Substituttion GroUtp)s of Degree Sixteen.

As the required substituitioni mtust be transfornmed into its 5th power by af. be. cg ih. kj. .l, it must be one of the tlhree used as factors ill the following products:

aebfcy injkml . dh. op? x ahcf. bedy. imkn .jl = accb . gl . jnlm. op fbg.ce inj'-?kIl . d. 01) x aljl edy inkj - = ed. elhgf .jnlrn . o,

agbecf. inj]km 1. Cdh . 0) X ahcf. beclg . imkns . jl abdc . eflhg .jnlm-. Op.

From these products it follows that only the last can occur in G since G' is of class 12. The transforms of its cube generate a self-conjugate subgroup of order 16.

The G' of order 720 may be genierated by the precedinig G' and ae. bf . cg. dhi. Since the squares of the first two of the giveni three products are liot contained in its subrgoup of order 48 generated by the first two given generators of the precedillg G and the one jtust givenl, ther-e can be only one G that contains this G'. It evidently contains the same self-conjugate subgroup of order 16 as the preceding G. These two groups are doubly tranisitive. As the remaining two primitive groups of degree 15 that do n1ot containl the alternating grotup are doubly transitive, the corresponding groups of degree 16 will be triply tranlsitive.

The G' of order 2520 may be geinerated by

aem .bfn .gjk. hi1,, cek dfl . gin .Jm, akodn. bfnij . cgeh1.

All the substitutions of G' that are similar to the first of these genlerators are conljugate in G' anid eaclh is transformed into itself by 9 substitutionis which per-

imtte 3 of its systemus accordinog to the alternating group. Hence G mnuLst contain oiie of the following substitutions:

abefmn)( 71,jikl 'cd. o1p afrnmb gijlkh7 co . dp aneblinf) gljhki J cl. do

Since the required substitution must be transformed inito its 5th power by the last of the three genlerators givenl above, it must be one of the three employed in forming the followinig products:

anebmnf . g4jiki . cd. op1 X akodn. bfrnij. cgehli = biopdglef kcnli,

anebjf . gijlkih. cd. op x akodn . b/inij. cgehI = bi. cn?lefklopdgj, anitebqmtf . gljkhi. cd. op x akodt . bf in. cgehl = biefkjl . cnhiopdg.



MILLER: On the Primitive S'ubstitutiotn Group)s of Degree Sixteen. 241

From these products it follows that only the last can occur in G. The transforms of its cube generate a self-conjugate subgroup of order 16.

The G' of oirder 20160 may be generated by

ab. cd .ef. gh, aen,. bfn. gjk .i1, ackegemj. bldlhfni, aocgjmk. bd?f1lih.

As all the substitutions of G' that are similar to the second of these generators aire conjugate, this G must also contain one of the 27 substitutions given in the preceding case. Sinice the required substitution must be transformed into its 5th

power by em .fn. gk. hl, which is contained in G', it must be one of the three used as factor in the following equations:

anebmf. gjflhki . cd. op x akcgemj. bldhfni= aiel. bjfk. ch. dg. mnn. 01X), anebmf . gljhlki. co . clp X akegenij. bld4ifni =aiel. bjfk. cogdph . mn, aqzebmf . ghkki . cp . do X akcgemj. bldbZfnzi aiel. bjfk. cpgdol. mn.

Hence only the first of these three can occur in a G. The transforms of its cube generate a self-conjugate subgroup of order 16. We have now considered all the possible primitive groups of degree 16 that do not contain the alternating group, and have found no group that is not contained in the enumeration of the first section. It may be observed that the substitutions uponi which otur argu- ments have been based may be selected in many different ways. As suitable substitutions can readily be found by means of the given generating substitutions, it did not seem necessary to indicate in every case how the particular one that has been employed has been obtained.

It is well known that a solvable primitive group must be of degree pn, p

being a primne number.* The following table gives the enumeration of all these groups whose degree is less than 27:

Degree, 3 4 5 7 8 9 11 13 16 17 19 23 Number of groups, 2 2 3 4 2 7 4 6 10 5 6 4

PARIS, June, 1897.

* Cf. Jordan, " Trait6 des Substitutions," p. 398.

32



on the primitive substitution groups of degree sixteen

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