206 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



coordinates are (0, 1, 0, 0, 1). For convenience of notation we shall gen- 

 erally omit commas in writing a sequence. The above 5-place sequence 

 will be written, for example, 01001. 



We define the product of two n-ylace hinarij sequences, aicii • • • a„ and 

 ^1^2 • ■ • bn as the n-place binary sequence 



fli + hi , a-i ■]- h-i , ■ ■ • , ttn + hn 



Here the a's and 6's are zero or one and the + sign means addition 

 modulo 2. (That is + 0=1 + 1 = 0, 0+1 = 1+0=1) 

 For example, (01101) (00111) = 01010. With this rule of multiplication 

 the 2" w-place binary sequences form an Abelian group of order 2". 

 The elements of the group, denoted by Ti , T'2 , • • • , Tin, say, are the 

 n-place binary sequences ; the identity element I is the sequence 000 • • • 

 and 



IT, = Til = T. ■ T,Tj = TjTr, TiiTjT,) = iTiTj)Tk ; 



the product of any number of elements is again an element; every ele- 

 ment is its own reciprocal, Ti = Tf^, TI = /. We denote this group 

 by Bn . 



All subgroups of Bn are of order 2 where k is an integer from the set 

 0, 1, 2, • • • , n. There are exactly 



N{n, k) = 



(2" - 2") (2" - 2') (2" - 2') • • • (2" - 2'-') 



(2^ - 2»)(2'^ - 20(2* - 22) 

 = N(n, n — k) 



{2" - 2'-') 



(2) 



distinct subgroups of Bn of order 2 . Some values of N(n, k) are given in 

 Table I. 



Table I — Some Values of A^(n, k), the Number of Subgroups 

 OF Bn OF Order 2''. N(n, k) = N{n, n — k) 



