关于群的一致覆盖的Herzog-Schönheim猜想
On the Herzog-Schönheim conjecture for uniform covers of groups
摘要：
设G是任意群，$a_1G_1,...,a_kG_k (k>1)$是G的左陪集.在1974年赫尔佐格和Schönheim证明，如果$\Cal A=\{a_iG_i\}_{i=1}^k$是一个划分的G然后（有限）指数$n_1=[G:G_1],...,n_k=[G:G_k]$不能不同。本文证明了：若$\Cal A$同时覆盖G的所有元素，且$G_1,...,G_k$是G的次正规子群，且不全等于G，则$M=\max_{1\le j\le k}|\{1\le i\le k:n_i=n_j\}|$不小于$n_1... n_k$的最小素除子，且$\min_{1\ls i\ls k}\log n_i=O(M\log^2 M)$的O-常数是绝对的.
英文摘要：
Let G be any group and $a_1G_1,...,a_kG_k (k>1)$ be left cosets in G. In 1974 Herzog and Schönheim conjectured that if $\Cal A=\{a_iG_i\}_{i=1}^k$ is a partition of G then the (finite) indices $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot be distinct. In this paper we show that if $\Cal A$ covers all the elements of G the same times and $G_1,...,G_k$ are subnormal subgroups of G not all equal to G, then $M=\max_{1\le j\le k}|\{1\le i\le k:n_i=n_j\}|$ is not less than the smallest prime divisor of $n_1... n_k$, moreover $\min_{1\ls i\ls k}\log n_i=O(M\log^2 M)$ where the O-constant is absolute.
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