Subnormal depth

Definition

The subnormal depth (also sometimes called the defect or subnormal defect) of a subnormal subgroup $H$ in a group $G$ is defined in the following equivalent ways:

• It is the smallest $n$ for which there exists an ascending chain of subgroups $H=H_{0}\leq H_{1}\leq H_{2}\leq \dots \leq H_{n}=G$ where each $H_{i}$ is normal in $H_{i+1}$.
• Consider the sequence $G_{0}=G$, $G_{i}$ is the normal closure of $H$ in $G_{i-1}$. The subnormal depth is the smallest $n$ for which $G_{n}=H$.
• Consider the sequence $K_{i}$ where $K_{0}=G$ and $K_{i+1}=[H,K_{i}]$. The subnormal depth is the smallest $n$ for which $K_{n}\leq H$.

We typically say that a subgroup has subnormal depth $k$ if its subnormal depth is less than or equal to $k$. A subgroup of subnormal depth (less than or equal to) $k$ is termed a $k$-subnormal subgroup.

Equivalence of definitions

Further information: Equivalence of definitions of subnormal subgroup

Particular cases

• A subgroup has subnormal depth $0$ if and only if it is the whole group.
• A subgroup has subnormal depth (less than or equal to) $1$ if and only if it is a normal subgroup.
• A subgroup has subnormal depth (less than or equal to) $2$ if and only if it is a 2-subnormal subgroup.
• A subgroup has subnormal depth (less than or equal to) $3$ if and only if it is a 3-subnormal subgroup.
• A subgroup has subnormal depth (less than or equal to) $4$ if and only if it is a 4-subnormal subgroup.