Ascending chain condition on subnormal subgroups implies subnormal join property

Statement
Any group satisfying ascending chain condition on subnormal subgroups, i.e., any group in which there is no infinite ascending chain of subnormal subgroups, also satisfies the following two conditions:


 * 1) It is a group satisfying subnormal join property, i.e., the join of any two (and hence finitely many) subnormal subgroups of the group is again subnormal.
 * 2) It is a group satisfying generalized subnormal join property, i.e., the join of any, possibly infinite, collection of subnormal subgroups of the group is again subnormal.

Applications

 * Noetherian implies subnormal join property
 * Finite implies subnormal join property

Stronger facts

 * Derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property

Similar facts

 * Nilpotent derived subgroup implies subnormal join property
 * Join of normal and subnormal implies subnormal of same depth
 * 2-subnormal implies join-transitively subnormal
 * Subnormality is normalizing join-closed
 * Subnormality is permuting join-closed

Facts used

 * 1) uses::Ascending chain condition on subnormal subgroups is normal subgroup-closed
 * 2) uses::Join-transitively subnormal of normal implies finite-conjugate-join-closed subnormal
 * 3) uses::Finite-conjugate-join-closed for a subgroup property in a group satisfying ascending chain condition on that subgroup property implies conjugate-join-closed for that subgroup property
 * 4) uses::Conjugate-join-closed subnormal implies join-transitively subnormal

Proof of subnormal join property
We prove the statement by induction on the subnormal depth of one of the two subgroups whose join we are taking. Specifically, we prove the following statement by induction on $$h$$, for $$h$$ a nonnegative integer.

Formulation to be proved by induction on $$h \ge 0$$: Any subgroup of subnormal depth $$h$$ in a group satisfying the ascending chain condition on subnormal subgroups is join-transitively subnormal. In long form: Suppose $$G$$ satisfies the ascending chain condition on subnormal subgroups, $$H$$ is a subnormal subgroup of $$G$$ of subnormal depth $$h$$, and $$K$$ is a subnormal subgroup of $$G$$ of subnormal depth $$k$$. Then, the join of subgroups $$\langle H, K \rangle$$ is also a subnormal subgroup of $$G$$.

Note that when we apply this induction, the ambient group for which we use the inductive hypothesis is not the same as the ambient group for which we want to prove the goal of the inductive step.

Base case for induction: This is the case $$h = 0$$. In this case, $$H = G$$, so $$\langle H, K \rangle = G$$ is also subnormal. This settles the base case.

Inductive step ($$h - 1 \to h$$):

Inductive hypothesis: In a group satisfying the ascending chain condition on subnormal subgroups, any subnormal subgroup of subnormal depth at most $$h - 1$$ is join-transitively subnormal. In other words, in any group satisfying ascending chain condition on subnormal subgroups, the join of any two subnormal subgroups is subnormal if either of them has subnormal depth at most $$h - 1$$.

Inductive goal:

Given: $$G$$ is a group satisfying the ascending chain condition on subnormal subgroups. $$H$$ is a subgroup of subnormal depth $$h$$ in $$G$$

To prove: $$H$$ is join-transitively subnormal in $$G$$.

Proof: The key proof idea is to use the inductive step within a smaller ambient group, namely the normal closure of the subgroup we are starting out with, and then bootstrap from this to the bigger group. There are two key places where the ascending chain condition on subnormal subgroups is used. The first is that this property is a normal subgroup-closed group property (Fact (1)), allowing the shift down to the normal closure. The second is that the property allows us to go from joins of finitely many conjugate subgroups to joins of infinitely many conjugate subgroups (Fact (3)).

Proof of generalized subnormal join property
This follows directly from the ascending chain condition and the subnormal join property.