Join of 3-subnormal and finite subnormal implies subnormal

This article describes a computation relating the result of the Join operator (?) on two known subgroup properties (i.e., 3-subnormal subgroup (?) and Finite subnormal subgroup (?)), to another known subgroup property (i.e., Subnormal subgroup (?))
View a complete list of join computations

Statement

Statement with symbols

Suppose $G$ is a group, $H$ is a 3-subnormal subgroup of $G$ and $K$ is a finite subnormal subgroup of $G$. Then, the join $\langle H,K\rangle$ is a subnormal subgroup of $G$.

Further, if $k$ is the subnormal depth of $K$ and $r$ is its order, the subnormal depth of $\langle H,K\rangle$ is at most:

$2^{r}k+1$

Facts used

1. 2-subnormal implies join-transitively subnormal: In fact, the join of a 2-subnormal subgroup and a $s$-subnormal subgroup is $2s$-subnormal.
2. Join of subnormal subgroups is subnormal iff their commutator is subnormal: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
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