Write cyclic groups of order $p^n$ in terms of simple groups Some say that studying simple groups helps you understand the structure of non-simple groups.
How can I write in terms of simple groups $\mathbb{Z}_{p^n}$?
Eg. $\mathbb{Z}_9$ 
 A: The group $\mathbb{Z}_{p^n}$ has a composition series
$$1\lhd \mathbb{Z}_{p}\lhd \mathbb{Z}_{p^2}\lhd\cdots\lhd\mathbb{Z}_{p^{n-1}}\lhd\mathbb{Z}_{p^{n}},$$
in which each of the factors $\mathbb{Z}_{p^{i}}/\mathbb{Z}_{p^{i-1}}$ is a simple group of order $p$. This can be easily proved by induction on $n$, using the facts that a $p$-group has non-trivial centre, factoring out by a central subgroup of order $p$, and then using the correspondence theorem to pull back a composition series from from the quotient to one for the original group.
More generally, any finite group has a composition series
$$1\lhd G_{1}\lhd G_{2}\lhd\cdots\lhd G_{n-1}\lhd G_{n},$$
in which each factor  $G_{i}/G_{i-1}$ is simple (perhaps a cyclic group of prime order).
The set of composition factors $G_{i}/G_{i-1}$ is uniquely determined by the group, up to isomorphism and permutation (Jordan–Hölder Theorem).
Beware that an infinite group (such as $\mathbb{Z}$) may fail to have a composition series.
The importance of this is that any group with a composition series (in particular, any finite group) can be built up, step-wise, from simple groups.  It is, in a sense, an analogue of the Fundamental Theorem of arithmetic, which says that any (positive) integer is (uniquely) a product of primes.  Here, the "primes" are the finite simple groups.  Unfortunately, the problem of determining all the ways in which a finite group can be constructed from simple groups in this way is quite difficult.  This is called the "extension problem".
To appreciate the difficulty, imagine that every group of order $1024$ has essentially the same composition factors (basically, $10$ copies of $\mathbb{Z}_{2}$), but there are $49487365422$ different groups of order $1024$, up to isomorphism!
