Transversal coset partitions of groups

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corollary_10: An index list is realized by a coset partition of some infinite group exactly when it is realized by one of some finite group, with properties kept under quotients and finite direct products carried along, so nilpotent groups satisfy Herzog–Schönheim.

theorem_2: For distinct proper subgroups H and K of any group G, a partition of G into left cosets of both H and K exists exactly when H and K do not generate G, and every such partition splits whole cosets of the join.

theorem_3: For three distinct, proper, mutually commuting subgroups H, K, L of a group G, a transversal coset partition exists exactly when G is not HKL, and every one arises by splitting HKL-cosets, then cosets of products of two, then cosets of single subgroups.

theorem_4: For distinct proper subgroups H, K, L of any group that do not all have index 3, no partition of the group uses exactly one coset of each, with no commutation hypothesis.

theorem_5: For any group with four mutually commuting distinct proper subgroups, one of index 2, no coset partition uses exactly one coset of each of the four.

theorem_6: Any partition of any group into cosets of two to seven distinct proper subgroups, using each of them, contains two cosets whose subgroups have the same index; the cases of five to seven subgroups rest on a computer search.

theorem_7: A finite direct product of at least four nontrivial subgroups has a partition into cosets of all its factors, and no standard construction yields it.

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Fusun Akman and Papa A. Sissokho, "Transversal coset partitions of groups," Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 66(2), 417--441, 2025. https://doi.org/10.1007/s13366-024-00748-9.

The copy read for this card is the author-typeset manuscript, without its ancillary program. The manuscript prints no journal header, arXiv stamp, copyright or license line, and its download URL was not recorded; no arXiv record exists for the paper (an arXiv title query on 2026-10-02 returned no result), and the version of record's publisher page for DOI 10.1007/s13366-024-00748-9 sits behind a login wall; the term is unstated.

For distinct subgroups $H_1,\ldots,H_r\leq G$, the paper calls a coset partition transversal when at least one $H_i$-coset occurs for every $i$, and pure when exactly one occurs for every $i$. Its Conjecture 1 says that if the $H_i$ are mutually commuting distinct proper subgroups, no pure transversal partition exists. The Herzog--Schönheim conjecture asks only that every partition of a group into $r$ cosets, $2\leq r<\infty$, have two cosets coming from subgroups of the same index; Conjecture 1 implies it when the subgroups mutually commute.

Main results bearing on distinct indices

The following statements are recorded with the source's hypotheses and conclusions. Page numbers refer to the manuscript's pages.

Theorem 2 (p. 2; proof in Section 3, p. 9). For any group $G$ and distinct proper subgroups $H,K$, an $\{H,K\}$-transversal coset partition exists if and only if $G\ne\langle H,K\rangle$. Whenever it exists, the $H$-cosets and $K$-cosets are obtained by fully decomposing selected left $\langle H,K\rangle$-cosets into $H$-cosets or $K$-cosets, respectively. Consequently Conjecture 1 holds even without $HK=KH$.

Theorem 3 (p. 2; proof in Section 4, pp. 10--12). If $H,K,L$ are distinct, proper, mutually commuting subgroups of $G$, an $\{H,K,L\}$-transversal coset partition exists if and only if $G\ne HKL$. Every such partition is obtained by decomposing $G$ into left $HKL$-cosets, then each of those fully into cosets of one of $HK,HL,KL$, and finally each of those fully into cosets of one of $H,K,L$. Hence Conjecture 1 holds. If none of $H,K,L$ is contained in the product of the other two, every such partition is a standard construction.

Theorem 4 (p. 3; proof in Section 5, p. 12). Let $H,K,L$ be distinct proper subgroups of any group $G$. If they do not all have common index $3$ in $G$, then no pure $\{H,K,L\}$-transversal coset partition exists; Conjecture 1 therefore holds here without mutual commutativity. The proof lists the only three unit-fraction possibilities as

$$[2,3,6],\qquad [2,4,4],\qquad [3,3,3],$$

and reduces the first two to the impossible pure two-subgroup case.

Theorem 5 (p. 3; proof in Section 5, p. 12). For any group $G$ with four mutually commuting distinct proper subgroups, one of which has index $2$ in $G$, Conjecture 1 holds. The one-sentence proof reduces to three mutually commuting subgroups, as in the proof of Theorem 4, and applies Theorem 3.

Theorem 6 (p. 3; proof in Section 5, pp. 12--13). For any group $G$, distinct proper subgroups $H_1,\ldots,H_r$, and $2\leq r\leq7$, every $\{H_1,\ldots,H_r\}$-transversal coset partition contains two distinct cosets $xH_i$ and $yH_j$, with $i=j$ allowed, such that $[G:H_i]=[G:H_j]$. The $r\leq4$ cases are eliminated from the unit-fraction and coprime-index restrictions, the $r=4$ list $[2,4,6,12]$ because the other three cosets would partition a coset of its index-$2$ subgroup, which after translation gives a partition of that subgroup with indices $[2,3,6]$. For $r=5$ the paper checks all $147$ decompositions of $1$ into five unit fractions, repetitions included, each of which has a coprime pair, a $2$ or a repetition, and then reports that a Haskell program verified all cases $2\leq r\leq7$ by enumerating distinct unit-fraction lists and eliminating those containing denominator $2$ or a coprime pair. Appendix A (pp. 20--21) prints the $r=3,4,5$ outputs, the lists with distinct denominators, and points to external code; that code was not read.

Thus a counterexample to Problem 274 in the distinct-index formulation must contain at least eight cosets belonging to at least eight distinct subgroups. This is only a lower bound: the computation stopped at $r=7$, and the paper gives numerical distinct-index candidates with $r=13$ and $r=15$ (the latter from its reference [20]) that survive its three elementary filters, without realizing either list as a group coset partition.

Theorem 7 (p. 3; construction in Section 6, pp. 13--16). If $r\geq4$ and the finite group $G=H_1\times\cdots\times H_r$ is a direct product of nontrivial subgroups, then $G$ has an $\{H_1,\ldots,H_r\}$-transversal coset partition, necessarily not obtainable by a standard construction. The proof chooses mutually disjoint product cosets $a_iH_iH_r$ for $1\leq i<r$, decomposes each into $H_i$-cosets, and decomposes the nonempty remainder into $H_r$-cosets.

Finite and infinite transfer

Corollary 10 (p. 4). For an index list $D=[d_1^{n_1},\ldots,d_r^{n_r}]$, representing $n_i$ cosets from each of $r$ distinct finite-index subgroups:

1. Some infinite group has a coset partition with index list $D$ exactly when some finite group has one.
2. Properties preserved by quotients and finite direct products, including nilpotence, solvability, and abelianness, can be carried between the finite and infinite realizations.
3. All nilpotent groups, and hence all abelian groups, satisfy the Herzog--Schönheim conjecture, by the cited finite nilpotent result [7].

The inputs are Lemma 8 (p. 4), which passes an infinite-group partition to a finite quotient without changing its index list, and Lemma 9 (p. 4), which forms product partitions and extends a finite example to an infinite one by multiplying by the one-part partition of an arbitrary infinite group. Corollary 11 (p. 5) gives the analogous finite/infinite equivalence while preserving mutual commutativity of the distinct subgroups.

This transfer is about finite subgroup indices, the paper's formulation of Herzog--Schönheim. For finite $G$, distinct subgroup indices are equivalent to distinct coset cardinalities. That distinction should remain explicit when reading the word "sizes" in Problem 274 for an infinite group.

Direct-product and parallelism constructions

The first concrete nonstandard construction is Example 15 (p. 7), expanded in Example 24 (pp. 13--14): for $G=H\times K\times L\times M$ with all four factors of order $2$, the eight cosets

$$cH,\ bdH,\ K,\ aK,\ dL,\ adL,\ bcM,\ abcM$$

partition $G$. Section 6 then generalizes this selection argument to Theorem 7. Remark 26 (p. 16) permits factors indexed by a partition $T_1\sqcup\cdots\sqcup T_m=\{1,\ldots,r\}$ with $m\geq4$, producing further iterations of standard and nonstandard constructions.

Proposition 27 (pp. 17--18). Write a transversal partition as

$$\mathcal P=\{a_{ij}H_i:1\leq i\leq r,\ 1\leq j\leq m_i\},$$

and suppose $[G:H_i]=rm_i$ for every $i$. If a subgroup $\Gamma=\{\gamma_1,\ldots,\gamma_r\}$ satisfies

1. $\Gamma H_i=H_i\Gamma$ for every $i$;
2. $\Gamma\cap H_i=\{1\}$ for every $i$;
3. $|\Gamma|=r$; and
4. $a_{ij}\Gamma H_i\cap a_{i\ell}H_i=\varnothing$ whenever $j\ne\ell$,

then

$$\mathcal P\Gamma =\{a_{ij}\gamma_kH_i:1\leq i,k\leq r,\ 1\leq j\leq m_i\}$$

is an exact $r$-cover and a coset parallelism: every coset of every $H_i$ occurs exactly once. If, in addition, each element of $\Gamma$ commutes with every representative $a_{ij}$, then $\mathcal P_k=\gamma_k\mathcal P$ for $1\leq k\leq r$ are transversal coset partitions and together form a transversal coset parallelism.

Corollary 31 (p. 19). Let $[G:H]=r$, let $N_G(H)=H$, and let the abelian subgroup $\Gamma=\{\gamma_1,\ldots,\gamma_r\}$ of order $r$ be complementary to $H$. Then:

1. $\Gamma$ is a full set of right-coset representatives for $H$;
2. the conjugates $H_i=\gamma_i^{-1}H\gamma_i$ are distinct and each is complemented by $\Gamma$;
3. in the source's exact wording, $\mathcal P=\{\gamma_1H_1,\ldots,\gamma_rH_r\}$ is a pure transversal coset partition "of $H$"; and
4. $\gamma_1\mathcal P,\ldots,\gamma_r\mathcal P$ form a transversal coset parallelism of $G$.

The proof invokes Lemma 12, while part (4) translates partitions of $G$, so the displayed domain in part (3) appears to be a typographical error for $G$. The construction is treated below according to that supported reading rather than as a partition of $H$ by ambient $G$-cosets.

Construction locators for Proposition 27 and Corollary 31 are Example 29 (p. 18), a transversal parallelism in $C_2\times C_2\times C_2\times C_3$; Example 30 (p. 18), the Klein-four translation group for Example 24; Example 32 (p. 19), the odd-index dihedral construction; and Example 33 (p. 19), the $S_4$ construction using its normal Klein four-group. The precursor pure partitions are Examples 13 and 14 (p. 6), obtained from cosets of conjugates by Lemma 12.

None of these explicit constructions answers Problem 274:

• Example 24 has two cosets from each order-$2$ factor, and all four factors have index $8$.
• In the general Theorem 7 construction, each $a_iH_iH_r$ decomposes into $|H_r|\geq2$ cosets of $H_i$, so an index is repeated before the remaining $H_r$-cosets are added.
• Example 29 uses three cosets apiece from subgroups $H,K,L$ of index $12$ and two cosets from $M$ of index $8$.
• Proposition 27 begins with multiplicities $m_i$: if some $m_i>1$, its index repeats within $\mathcal P$; if every $m_i=1$, its equation $[G:H_i]=rm_i$ makes all indices equal to $r$. The combined $\mathcal P\Gamma$ is moreover an exact $r$-cover, not an exact one-cover, though its translated components are partitions under the extra hypothesis.
• Corollary 31 and Examples 13, 14, 32, and 33 are pure, but their subgroups are conjugate and all have the same index $r$.

Reading and verification status

Read status: claims checked. The manuscript was read end to end. The hypotheses and conclusions of Theorems 2--7, Lemmas 8 and 9, Corollary 10, Proposition 27, and Corollary 31 were checked clause by clause, together with the construction passages and examples located above. Their proofs were read for the stated construction and obstruction summaries but have not been independently verified. The Haskell computation reported for Theorem 6 was not replayed because its ancillary code was not read.

Results.

• Theorem 2 (p. 2): two distinct proper subgroups $H,K$ admit a transversal coset partition exactly when $G\ne\langle H,K\rangle$; never a pure one.
• Theorem 3 (p. 2): the structure of transversal partitions for three mutually commuting subgroups; never a pure one.
• Theorem 4 (p. 3): no pure partition by three distinct proper subgroups whose indices are not all $3$.
• Theorem 5 (p. 3): no pure partition by four mutually commuting subgroups when one has index $2$.
• Theorem 6 (p. 3): with two to seven distinct proper subgroups, some index repeats.
• Theorem 7 (p. 3): nonstandard transversal partitions of finite direct products of $r\geq4$ nontrivial factors.
• Corollary 10 (p. 4): finite and infinite groups realize the same index lists, with Lemmas 8 and 9.

Bears on. Problem 274: Theorem 6 excludes a partition of any group into two to seven cosets of pairwise different indices, so a counterexample in that formulation has at least eight cells; Theorems 2 and 4 are its cases of two and three cells without the computer search, and Theorem 5 the four-cell case for mutually commuting subgroups one of which has index $2$. Corollary 10 shows that an index list is realized in some group exactly when it is realized in a finite group, and records that nilpotent groups satisfy the conjecture by Berger, Felzenbaum and Fraenkel. Theorem 7 and the parallelism constructions all repeat an index, so none of them gives a partition of the kind the problem asks for.

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