A181566 - OEIS

%I #17 Oct 09 2023 13:06:49

%S 0,0,0,2,0,0,0,4,3,0,0,3,0,0,0,8,0,3,0,4,0,0,0,7,5,0,8,5,0,0,0,16,0,0,

%T 0,9,0,0,0,12,0,0,0,5,4,0,0,19,7,5,0,6,0,12,0,16,0,0,0,6,0,0,5,37,0,0,

%U 0,6,0,0,0,20,0,0,5,6,0,0,0,32,27,0,0,6,0,0,0,26,0,5,0,7,0,0,0

%N Minimum number of random elements such that their orders allow identification of an abelian group of order n (sampled uniformly) with probability greater than 1/2.

%e For n=4, by the fundamental theorem of finite abelian groups, the group is either Z4 or Z2 x Z2. When you choose 2 random elements, if 1 element comes out of the 2 elements of order 4, you will know you have Z4. If the 2 elements are of order 2 in Z2 x Z2, you will know you have Z2 x Z2. Calculating the probabilities, when you choose 2 random elements, if the group is Z4, there is a 5/6 chance of knowing it. If it is Z2 x Z2, there is a 1/2 chance of knowing it. Since we assume each non-isomorphic abelian group of order n has the same probability of being the group, averaging 5/6 and 1/2 we get a 2/3 chance that the group is known after choosing 2 elements. Since the probability that a single random element will allow us to identify the group is 1/4, which is not greater than 1/2, a(4) = 2.

%Y Cf. A181189.

%K nonn

%O 1,4

%A _Isaac Lambert_, Oct 30 2010

%E Edited and terms a(16) onward added by _Max Alekseyev_, Oct 07 2023