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J,0 (-1)ib-1 2 ' if lb) > 1. 2). 2 is absolutely convergent and alternating, the theorem follows by grouping the terms in pairs. 1 SOME APPLICATIONS Generalized elementary Abelian groups. In this and the following two sections we shall collect the applications indicated above, establish some new ones, and make conjectures regarding others. 6 we noted that when b is a prime N(b) is the number of subgroups of order bg in an elementary Abelian group of order bk. 3 which established that 2(b) represents the number of subgroups in an elementary Abelian group of order bk when b is a prime.

1t The two theorems of this section completely characterize 1t(b). 1. 1b) fails to exist if and only if (1) b = 0 and t is ( not a non-negative integer, or (2) b is a root of unity different from 1 and t is not an integer. PROOF. 91. 2). 2. PROOF. 2]. 4. Some results on In this section we will establish a formula for 0-01) when Ibl > 1. 1. PROOF. If t is a non-negative integer, the theorem follows from If Ibl > 1, then Otb) exists for all t. 2). 2 and the ratio test. 2. O) PROOF. b-jne+J-1(b) j=0 (b) if Ibl >1.

Algebra. 3. 07n (1-b-j)-1 - j=1 kb) if Ibl > 1. 2, (b) 0-(1+1) (t) ( b-i i ][1 b-(11+1)]-1. 2) reduces to 2 b-j(n+1 )Nn+j-1 (b). 3, 28 0 -(n+1) (b) b-3h+1)Nn+3-1(t). 0-0(3r)1+1). 1) now follows, and the proof by finite induction is complete. 5. Some results on (b). 1. When t is not a non-negative integer, 2(t) exists if 1 and does not exist if Ibl lb' <1. 2. PROOF. 50. < b- PROOF. 2-1 (b) . j,0 (-1)ib-1 2 ' if lb) > 1. 2). 2 is absolutely convergent and alternating, the theorem follows by grouping the terms in pairs.

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