Dear all,
For the additive group Z of all integers, its subgroups other than {0}
have the form nZ and a coset of nZ is just a residue class a+nZ. If
n=p_1...p_r where p_1,...,p_r are (not necessarily distinct) are
primes, then we define
f(n)=(p_1-1)+...+(p_r-1).
The function f is called the Mycielski function.
I have the following conjecture.
CONJECTURE. Let a_1G_1,...,a_kG_k be finitely many left cosets in a
group G. Suppose that {a_sG_s}_{s=1}^k covers each element of G at
least m>0 times but {a_sG_s}_{s not=t} does not. Then k is at least
m+f([G:G_t]) and hence [G:G_t] is at most 2^{k-m}. (It is known that
[G:G_t] must be finite.)
In my paper with G. Lettl ["On covers of abelian groups by cosets",
Acta Arith. 131(2008), no. 4, 341-350] we proved that the conjecture
holds when G is abelian.
In my paper "Exact m-covers of groups by cosets" [European
J. Combin. 22(2001), no.3, 415-429] I extended a result of Berger,
Felzenbaum and Fraenkel by showing that if {a_sG_s}_{s=1}^k covers
each element of G exactly m times and G/(G_t)_G is solvable then k is
at least m+f([G:G_t]). The reader may also consult my paper "Finite
covers of groups by coesets or subgroups"
[Internat. J. Math. 17(2006), no.9, 1047-1064] for other related
results. These papers and my related talks are available from my
homepage
http://math.nju.edu.cn/~zwsunIt seems that my above conjecture is very challenging and its proof
might involve the representation theory of finte groups.
Best wishes
Zhi-Wei Sun (Nanjing University, P. R. China)
