My professor gave the above definition for a series of modules.
I have a few questions:
1) He defined $n$ to be the length of the series. But then he proceeds to say that $n$ equals the number of factors. However, given that he has not imposed any conditions on the submodules apart from the fact that they are subsets, its possible that some of the factors could be the same. Does that mean that $n$ includes multiplicities i.e. same factors or possibly isomorphic factors i.e the factors are different based solely on the indices of the submodules.
($\frac{M_{i+1}}{M_i}=\frac{M_{j+1}}{M_j} j\neq i$ even though both factors could be same isomorphically i.e. $0$ quotient modules.)
2) My second question relates to the first. His definition of equivalence includes a bijection between the sets of factors. Hence, in a set of factors are the factors distinguished solely on the indices of the submodules as described above or are multiplicities not allowed.
First, often we consider the factors as a sequence $\frac{M_1}{M_0},\frac{M_2}{M_1},\ldots,\frac{M_n}{M_{n-1}}$ rather than a set, which completely bypasses the possible problem of multiplicities. In this context, an equivalence of two series is given by a permutation of $\{1,2,\ldots,n\}$ such that the permuted factor sequence is isomorphic, element-wise, with the other factor sequence.
Not much is lost if we consider them as sets, though, since the only time we have true duplicates is when $M_{i-1} = M_i = M_{i+1}$. When we talk about the set of factors modulo isomorphism, however, yes, we need to count multiplicities.