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Source. Theorem 9', printed p. 48, physical PDF p. 7 of the retained image scan.

Statement

Let $H$ be a subnormal subgroup of finite index in a group $G$. The least positive integer $k$ admitting elements $a_1,\ldots,a_k\in G$ and subnormal subgroups $G_1,\ldots,G_k$ of $G$ such that

$$G=a_1G_1\mathbin{\dot\cup}\cdots\mathbin{\dot\cup}a_kG_k \qquad\text{and}\qquad \bigcap_{i=1}^kG_i=H$$

is $1+d(G,H)$.

Here the dotted union records the source's phrase "left coset decomposition": the cosets form a partition of $G$.

Proof pointer. The source obtains the statement by combining Theorems 4 and 8'; Theorem 8' and its proof run across printed pp. 47--48. That proof was not reconstructed or independently checked here.

Bears on. Qualified structural context for Problem 274. The theorem does not assert that the cosets have different sizes.