- #1

Kreizhn

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## Homework Statement

Let G be a finite cyclic group and [itex] \ell(G) [/itex] be the composition length of G (that is, the length of a maximal composition series for G). Compute [itex] \ell(G) [/itex] in terms of |G|. Extend this to all finite solvable groups.

## The Attempt at a Solution

Decompose |G| into its prime factors, say

[tex] |G| = p_1^{r_1} \cdots p_n^{r_n}. [/tex]

Since G is cyclic, it has a subgroup of order corresponding to every divisor of |G|, and this subgroup is normal. Further, since quotients of cyclic groups are cyclic, in order to ensure that the quotients are simple we need to give them prime order. Consequently, the maximal composition series is given by

[tex] \ell(G) = \sum_{i=1}^n r_i [/tex]

since for each step, we choose a prime [itex] p_{i_j} [/itex] and create the normal subgroup of order [itex] p_1^{r_1} \cdots p_{i_j}^{r_{i_j}-1} \cdots p_n^{r_n} [/itex] which then has prime index making its quotient simple.

Now problem one: It's unclear to me how to write [itex] \ell(G) [/itex] in terms of |G|. I feel like the totient should be involved somewhere.

Problem two: I'm not too sure how to extend this to finite solvable groups. I can show things like the group is solvable if and only if all composition factors (the quotients of the comp series) are cyclic. It seems to me that I could probably use this, though it's not clear to me how.