Subnormality is normalizing join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) satisfying a subgroup metaproperty (i.e., normalizing join-closed subgroup property)
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Statement

Suppose $H,K \le G$ are subnormal subgroups, with the property that $K \le N_G(H)$: in other words, $K$ normalizes $H$. Then the join of subgroups $\langle H, K \rangle$ is also subnormal. Moreover, the subnormal depth of $\langle H, K \rangle$ is bounded from above by the products of subnormal depths of $H$ and $K$.

Facts used

1. Join of normal and subnormal implies subnormal of same depth: If $L$ is normal in $G$ and $K$ is $k$-subnormal in $G$, then $KL$ is subnormal in $G$ with subnormal depth at most $k$.
2. Normality is upper join-closed: If a subgroup is normal in two intermediate subgroups, it is normal in their join.
3. Subnormality satisfies intermediate subgroup condition: More specifically, if $A \le B \le G$ are groups such that $A$ is $k$-subnormal in $G$, then $A$ is also $k$-subnormal in $B$.
4. Subnormal subgroup has a unique fastest descending subnormal series, where the series members are obtained by taking successive normal closures.

Proof

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Given: A group $G$, subnormal subgroups $H, K \le G$ such that $K \le N_G(H)$, i.e., $K$ normalizes $H$. $H$ has subnormal depth $h$ and $K$ has subnormal depth $k$.

To prove: $HK = \langle H, K \rangle$ is a subnormal subgroup, with subnormal depth at most $hk$.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 Consider the descending chain $G_i$ defined by $G_0 = G$, and $G_{i+1}$ is the normal closure of $H$ in $G_i$. This is the fastest descending subnormal series for $H$, and thus, $G_h = H$. Fact (4) $H$ is $h$-subnormal in $G$
2 $K$ normalizes $G_i$ for all $i$. In particular, for any $i$, $\langle G_i, K \rangle = G_iK$. $K$ normalizes $H$ Step (1) Any subgroup of $G$ defined deterministically in terms of $H$ must be invariant under any automorphism that leaves $H$ invariant.
3 For each $i$, $G_{i+1}$ is normal in $G_iK$. Fact (2) Steps (1), (2) By construction, $G_{i+1}$ is normal in $G_i$, and as observed in Step (2), $K$ normalizes $G_{i+1}$, so $G_{i+1}$ is normal in $G_iK$ (fact (2)).
4 For each $i$, $K$ is $k$-subnormal in $G_iK$ Fact (3) $K$ is $k$-subnormal in $G$ Given-fact combination direct
5 For each $i$, $G_{i+1}K$ is $k$-subnormal in $G_iK$ Fact (1) Steps (3), (4) Step-fact combination direct
6 $HK$ is $hk$-subnormal in $G$ Steps (1), (5) We have a chain: $HK = G_hK \le G_{h-1}K \le \dots \le G_1K \le G_0K = G$
where each member is $k$-subnormal in its successor. This tells us that $HK$ is $hk$-subnormal in $G$.