---

Source. Theorem 6, printed pp. 42--43, physical PDF p. 4 of the retained image scan.

Statement

Let $G$ be a group and $H$ a subnormal subgroup of finite index. If

$$[G:H]=\prod_{i=1}^r p_i^{\alpha_i}, \qquad f([G:H])=\sum_{i=1}^r\alpha_i(p_i-1),$$

and

$$d(G,H)=\sum_{j=1}^s([H_j:H_{j-1}]-1)$$

along a maximal chain $H=H_0\triangleleft H_1\triangleleft\cdots\triangleleft H_s=G$, then

$$[G:H]-1\geq d(G,H)\geq f([G:H])\geq\log_2[G:H].$$

Proof pointer. The proof is printed with the theorem on pp. 42--43. It uses the prime factorization of the successive indices in the chain. The proof was not reconstructed or independently checked here.

Bears on. This is an input to the source's coset-partition bounds and gives qualified context for Problem 274.