MR T. B. SPRAGUE ON A NEW ALGEBRA. 411 



Let a x = 5 .: a 4 = 2 



« 2 = 1 .'. a 3 = 4 

 a 5 = 6 .•. a 6 = 3 



The permutation is therefore 514263 ; and we see that 



s 3 (514263) = (241536); 

 whence *V(514263) = (362415) = ^(514263) . 



As a final example I will prove that, if a permutation is self- conjugate, there are (n — 1) 

 other self -conjugate permutations in the same set ; and every other permutation in the 

 set is conjugate to some other permutation in the set. 



If P is self-conjugate, we have P=_pP. Hence sVP = s h t r pV = psH h I* ; or the per- 

 mutation sY'P, which may represent any permutation in the same set as P, is conjugate 

 to s¥P. 



When h = k, we see that sYP is conjugate to sYP ; or, in other words, is self- 

 conjugate; and since we may give h any of the values 1, 2, . . . n— 1, there are n-l 

 self-conjugate permutations in the set besides P. 



For instance, the permutation P = (1756342) is self-conjugate ; and we have 



•s£P = (6452317) 

 sW = (3412765) 

 sH 3 ¥ = (3716542) 

 s^P = (6754312) 

 ,s 5 £ 5 P = (6432715) 

 sOT = (3216745) 



and we see that each of these is self-conjugate. Taking any other permutation in the 

 same set, say s 2 £ 5 P = (2765341), we see that this is conjugate to s 5 £ 2 P = (7156432). 



VOL. XXXVII. PART II. (NO. 19). 3 P 



