220 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



It is a considerable saving in notation in dealing with C„ to omit the 

 symbol "a" and write only the subscripts. In this notation for example, 

 the elements of d are 7, 1, 2, 3, 4, 12, 13, 14, 23, 24, 34, 123, 124, 134, 

 234, 1234. The product of two or more elements of C„ can readily be 

 written down. Its symbol consists of those numerals that occur an odd 

 number of times in the collection of numerals that comprise the sym- 

 bols of the factors. Thus, (12)(234)(123) = 24. 



The isomorphism between Cn and Bn can be established in many ways. 

 The most convenient way, perhaps, is to associate with the element 

 iii-2H ■ ■ ■ ik of Cn the element of Bn that has ones in places ii ,1-2, • • • , ik 

 and zeros in the remaining n — k places. For example, one can associate 

 124 of C4 with 1101 of Bi ; 14 with 1001, etc. In fact, the numeral no- 

 tation afforded by this isomorphism is a much neater notation for Bn 

 than is afforded by the awkward strings of zeros and ones. There are, 

 of course, other ways in which elements of C„ can be paired with elements 

 of Bn so that group multiplication is preserved. The collection of all such 

 "pairings" makes up the group of automorphisms of C„ . This group of 

 automorphisms of Cn is isomorphic with the group of non-singular linear 

 homogenous transformations in a field of characteristic 2. 



An element T of C„ is said to be dependent upon the set of elements 

 Ti , T2 , • • ■ , Tj oi Cn if T can be expressed as a product of some ele- 

 ments of the set Ti , T2 , • • • , Tj ; otherwise, T is said to be independent 

 of the set. A set of elements is said to be independent if no member can 

 be expressed solely in terms of the other members of the set. For example, 

 in Cs , 1, 2, 3, 4 form a set of independent elements as do likewise 2357, 

 12357, 14. However, 135 depends upon 145, 3457, 57 since 135 = 

 (145) (3457) (57). Clearly any set of n independent elements of Cn can 

 be taken as generators for the group. For example, all possible products 

 formed of 12, 123, and 23 yield the elements of C3 . 



Any k independent elements of C„ serve as generators for a subgroup 

 of order 2*". The subgroup so generated is clearly isomorphic with Ck ■ 

 All subgroups of C„ of order 2'' can be obtained in this way. 



The number of ways in which k independent elements can be chosen 

 from the 2" elements of C„ is 



F{n, k) - (2" - 2'')(2" - 2')(2" - 2') • • • (2" - 2'-') 



For, the first element can be chosen in 2" — 1 ways (the identity cannot 

 be included in a non-trivial set of independent elements) and the second 

 element can be chosen in 2" — 2 ways. These two elements determine a 

 subgroup of order 2\ The third element can be chosen as any element of 

 the remaining 2" — 2" elements. The 3 elements chosen determine a 



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