Finite Coverings of Groups

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corollary_2: Bounds the number of cosets in a group partition using the least common multiple of the subgroup indices.

theorem_6: Bounds the chain distance of a finite-index subnormal subgroup by its index and the Mycielski function.

theorem_9_prime: Characterizes one plus the subnormal distance as the least size of a coset partition with a prescribed subgroup intersection.

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Zhi-Wei Sun, Finite coverings of groups, Fundamenta Mathematicae 134 (1990), no. 1, 37--53, DOI 10.4064/fm-134-1-37-53.

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For

$$n=\prod_{i=1}^r p_i^{\alpha_i},$$

the paper uses Mycielski's arithmetic function

$$f(n)=\sum_{i=1}^r\alpha_i(p_i-1).$$

For a finite-index subnormal subgroup $H\leq G$, choose a maximal chain

$$H=H_0\triangleleft H_1\triangleleft\cdots\triangleleft H_s=G$$

and define

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

The source notes that this value is independent of the chosen chain. Theorem 6 compares $d(G,H)$ with the index and $f$. Theorem 9' characterizes $1+d(G,H)$ as the least size of a subnormal-coset partition with intersection $H$. Corollary 2 then gives an index-based lower bound for any such partition.

These statements give qualified structural context for Problem 274. They neither require nor produce pairwise different coset sizes, so they do not resolve that problem's exact question.

Compiled scope

The source identity and definitions on printed pp. 37 and 41--42, Theorems 6 and 7 on printed pp. 42--43, Theorem 9' on printed p. 48, and Corollary 2 on printed p. 49 were read from the image scan. The three selected statements below were transcribed. Their proofs and the paper's more general weighted covering theorem were not reconstructed or independently checked.

Results

• Theorem 6
• Theorem 9'
• Corollary 2

Bears on. Qualified context for Problem 274.