6 
MATHEMATICS: G. A. MILLER 
the 15. Thus every triad has an odd number, 3 or 1, of the odd set 
of 15, and an even number, 0 or 2, of the even set of 16. Any one of the 
odd set is found therefore in triads with 8 pairs from the even set, and 
these pairs can be arranged in 15 columns, an 8 by 15 array. Every odd 
element is found also with 7 pairs from the odd set. This leads to the 
tabulation of 15 columns of 7 pairs each, ranged above the columns of 
the 8 by 15 array. Every column is marked then by one odd element 
above it; the upper partial columns exhibit the head, or Ai5. 
Head and array form a convenient mode for constructing Asi's that 
are to have odd-and-even structure. If the head, the A15, is itself head- 
less, this tabulation is unique for that A31. I study here exclusively these 
odd-and-even Asi'^ whose head is a headless A15. Given any one such A31, 
tabulated, many others can be obtained by shifting the columns of its 
8x15 array while the head is kept stationary. To apply this method 
and to count the distinct Asi's that will be produced, one must know the 
groups G^f and G^', belonging to the head and to array respectively. 
The number of resulting Aai'^ is certainly not less than 15! divided by the 
product, d d\ of the orders of the groups belonging to the head and to the 
array respectively. These orders are small, whence the resulting A31 s are 
very many. 
Incidentally, if d and d' are relative primes, the resulting Asi's must 
be of the peculiar kind having no automorphic substitutions; i.e., their 
group is reduced to the identity. Such cases occur, e.g., with d = 2 and 
d' = Full details are to appear in the Transactions of the American 
Mathematical Society for January, 1915. 
THE ^>-SUBGROUP OF A GROUP OF FINITE ORDER 
By G. A. Miller 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF ILLINOIS 
Presented to the Academy, November 19. 1914 
A set of X operators ^i,^2r-5'^x of a finite group G is called a set of 
generators of G provided there is no subgroup in G which includes each 
of these operators. When these operators satisfy the additional con- 
dition that G can be generated by no X - 1 of them the set is said to be a 
set of independent generators of G. Those operators of G which can 
appear in none of its possible sets of independent generators constitute 
a characteristic subgroup, which was called by G. Frattini the (f)-sub- 
group of G. See Rend. Acc. Lincei, ser. 4, 1, 281 (1885). 
