Derived subgroup not is divisibility-closed
This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., derived subgroup) does not always satisfy a particular subgroup property (i.e., divisibility-closed subgroup)
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It is possible to have a group and a prime number such that is a -divisible group but the derived subgroup is not a -divisible group. In other words, for any , there exists such that , but there exists some for which there is no satisfying .
In fact, for any prime number , we can choose an example group specific to that prime. In fact, we can choose examples where is a divisible group for all primes, but the derived subgroup is not divisible by a specific prime of interest.
- General linear group over algebraically closed field is divisible
- Derived subgroup of general linear group is special linear group
- Special linear group over algebraically closed field is divisible precisely by those primes that do not divide its degree
For a single prime
Let be an algebraically closed field of characteristic zero. We could take for concreteness. Let .
By Fact (1), is divisible by all primes. In particular, it is divisible by .
By Fact (2), the derived subgroup of is .
By Fact (3), is not -divisible.
For a collection of primes
As with the previous example, let be an algebraically closed field of characteristic zero, such as .
Suppose is a finite collection of primes. We can construct an example of a group that is -divisible for all , but such that the derived subgroup is not -divisible for any . The idea is to take as the product of all the primes in , and set .
Fact (1) gives that is -divisible.
Fact (2) gives that the derived subgroup of is .
Fact (3) gives that is not -divisible for any .