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L(s)
Let G be an abelian group. L(s) denotes, for every z in B(G), the set of lengths of z, i.e., L(s) := {0} if s is the empty sequence (the identity of B(G)), and L(s) := {k \in N_+: s = a_1 * ... * a_k for some minimal zero-sum sequences a_1, ..., a_k \in B(G)} otherwise. For detailed notations, see book .
Then conjecture "|L(s^{k exp(G)})|=k|L(s^{exp (G)})-k+1|" is false by following example.

Let G = C_2 + C_2 + C_2 , s=a_1 a_2 ... a_7, where a_i's are all the seven different non-zero elements in G. Then
L(s)={2},
L(s^2)={4 5 6 7},
L(s^3)={6 7 8 9},
L(s^4)={7 8 9 10 11 12 13 14}.

|L(s^4)|=7:
( 1 0 0 )
( 0 1 0 )
( 0 0 1 )
( 1 1 1 )
~~~~~~~~
( 1 0 0 )
( 0 1 0 )
( 1 0 1 )
( 0 1 1 )
~~~~~~~~
( 1 0 0 )
( 1 1 0 )
( 0 0 1 )
( 0 1 1 )
~~~~~~~~
( 1 0 0 )
( 1 1 0 )
( 1 0 1 )
( 1 1 1 )
~~~~~~~~
( 0 1 0 )
( 1 1 0 )
( 0 0 1 )
( 1 0 1 )
~~~~~~~~
( 0 1 0 )
( 1 1 0 )
( 0 1 1 )
( 1 1 1 )
~~~~~~~~
( 0 0 1 )
( 1 0 1 )
( 0 1 1 )
( 1 1 1 )

 Halter-Koch, F., & Geroldinger, A. (2006). Non-unique factorizations: Algebraic, combinatorial and analytic theory. Chapman and Hall/CRC.
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