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Statement

A partition $\{g_iU_i\}_{i=1}^n$ of a group $G$ into cosets of subgroups of finite index admits multiplicity if $[G:U_i]=[G:U_j]$ for some distinct $i,j$ (p. 3).

Lemma 2.3 (p. 3). Let $\{g_iU_i\}_{i=1}^n$ be a coset partition of a group $G$ without multiplicity, and put $a_i=[G:U_i]$ for $1\le i\le n$. Then

• a) $a_i\ne a_j$ for distinct $1\le i,j\le n$;
• b) $\sum_{i=1}^n 1/a_i=1$;
• c) $\gcd(a_i,a_j)>1$ "for any $1\le i,j\le n$";
• d) if $G$ is a counterexample to the Herzog-Schönheim conjecture of minimal order, then $a_i>2$ for every $1\le i\le n$.

Part c) is printed for any $i,j$; the paper's restatement just below (p. 3) reads it for distinct pairs, and with $i=j$ it holds only when $a_i>1$, so it fails for the one-coset partition $\{G\}$, which is not a partition the conjecture concerns.

Source. L. Margolis and O. Schnabel, The Herzog-Schönheim conjecture for small groups and harmonic subgroups, Beitr. Algebra Geom. 60 (2019), no. 3, 399--418, doi:10.1007/s13366-018-0419-1. Labels and pages are those of arXiv:1803.03569v1, the edition the source card names: the lemma is on p. 3.

Read depth. Claims checked: the statement was read clause by clause against the print. The paper gives no proof beyond its sources; nothing here is independently reviewed.

Proof pointer

p. 3. The paper calls a) and b) clear: a) restates the absence of multiplicity, and b) records that each coset $g_iU_i$ takes the share $1/a_i$ of the group. Part c) follows from Lemma 2.2 (p. 3, after Ginosar and Schnabel [GS11, Corollary 2.1]): if $UV=G$ then no coset of $U$ is disjoint from a coset of $V$, and subgroups of coprime indices $r,s$ satisfy $UV=G$. Part d) is [GS11, Lemma 2.3].

Dependencies

Y. Ginosar and O. Schnabel, Prime factorization conditions providing multiplicities in coset partitions of groups, J. Comb. Number Theory 3 (2011), no. 2, 75--86 (the paper's [GS11]), Corollary 2.1 and Lemma 2.3.

Bears on

• Problem 274: for a finite group, an exact covering by two or more cosets of pairwise different sizes is a partition without multiplicity, so its indices satisfy a)--c), and in a counterexample of least order also d). These are the conditions from which the proof of Theorem A starts; they do not decide the problem.