Xon-regular transitive substitulion groups. 



71 



G^2 



amdiglchfkbnej 

 anflcljhhfjmekci 

 all . hjeicl . dnfkgm 

 aidhhli . cmejfl . gn 

 ajghcn . hldiek .fm 

 akcJidj . hmglfn . el 

 alfJiem . hncigj . dk 

 ambhf'i . cj . dlgken 

 anehgl . hi . ckfjdm 

 all . blciej . dmgkfn 

 akbhdi . clfjem . gn 

 anchgj . hkeidl .fm 

 ajdhck . hnf'igm . el 

 amehfl . hjgicn . dk 

 aifhhm . cj . dnekgl 

 alglien . hi . cmdjfk 



am . bl . ck . dj . ei .fh . gn 

 an . bm . cl . dk . ej .fi . gh 

 all . bwenck . diflgj 

 aigkcl . hngjeli .fm 

 ajfnei . bhcmgl . dk 

 akejgm . bi . cnfhdl 

 aldmhj . chekfi .gn 

 amcidn . bkghfj . el 

 anblfk . cj . dligiem 

 ah . bkcnem . djgJfi 

 akflbn . cj . dmeicjli 

 andicm . hjfhgk . el 

 ajbmdl . cifkeh . gn 

 amgjek . bi . cldhfn 

 aienfj . blgmcli . dk 

 alckgi . bhejdn .fm 



§ 3. 



The transitive groups of degree pq and of order pqr. 



The invariant subgroup of order 2^^ must be transitive. If it 

 is non-cyclical the largest group {H^) that is commutative to it 

 must be of order jj^g [p — 1). We can readily prove that the 

 Order of H^ does not exceed p'-q {p — 1), for H^ cannot transform 

 a Substitution of order q in the given invariant subgroup {Ho) 

 into more than jj positions. Another Substitution belonging to 

 the same division of II2 with respeet to its invariant subgroup 

 of Order p can then be transformed into no more than p) — 1 

 positions. As there are just pq substitutions that are commutative 

 to all the substitutions of Ho^) and the two given substitutions 

 generate H2 the given statement is proved. 



It is also easy to see that the order of H^ cannot be less 

 than p-q (p — 1), for the substitutions of order p which are 

 commutative to all the substitutions of T/o combined with B.j 

 generate a group of order p>^1- If we combine with this group 

 a Substitution of order p — 1 which transforms the substitutions 



'j Jordan, Traite des Substitutions, § 7.5. 



