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 $H = H_0 \le H_1 \le H_2 \le \dots \le H_n = G$ 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 $K_n \le H$.

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.