Subnormal depth

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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.