Composition length
This article defines an arithmetic function on a restricted class of groups, namely: group of finite composition lengths
Contents
Definition
Symbol-free definition
The composition length of a group is defined as the length of any composition series of the group. The composition length is well-defined only for a group of finite composition length.
By the Jordan-Holder theorem, the composition length is well-defined, because all composition series have the same length and in fact the same collection of composition factors.
Definition with symbols
Suppose is a group with a composition series:
.
Then, the composition length of is defined to be equal to .
Facts
Group property or condition | Meaning | What we can say about the composition length |
---|---|---|
trivial group | group of order one, has only the identity element | it is 0. In fact, the trivial group is the only group of composition length 0. |
simple group | no proper nontrivial normal subgroup | it is 1. In fact, a group is simple if and only if it has composition length 1. |
finite group | the underlying set is finite | The composition length is finite, and is bounded by the sum of the exponents on all prime divisors in a prime factorization. |
finite solvable group | finite and solvable, so in particular all the composition factors are groups of prime order. | The composition length equals the sum of the exponents on all prime divisors in a prime factorization. Specifically, if the order is with distinct primes, then the composition length is . In fact, the composition length is if and only if the group is solvable. |
group of prime power order | order is a prime power, i.e., of the form , prime. | composition length is , i.e., the prime-base logarithm of order. |
Relation with product notions
Product notion | Information for factors or inputs to the product | Result for product notion | Proof |
---|---|---|---|
external direct product | has composition length , has composition length | has composition length | composition length of direct product is sum of composition lengths |
external semidirect product | has composition length , and has composition length , with an action | has composition length | composition length of semidirect product is sum of composition lengths |
group extension | has composition length , has composition length | has composition length | composition length of extension group is sum of composition lengths |
wreath product | has composition length , has composition length on a set of size and its permutation action is on a set of size | has composition length |