Dear all,
For the additive group Z of all integers, its subgroups other than
{0} have the form nZ (n>0) 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)
primes, then we define
f(n)=(p_1-1)+...+(p_r-1).
In addition we set f(1)=0. The function f is called the Mycielski function.
I have the following conjecture for covers of general groups.
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 all the elements of G at
least m>0 times, with a_tG_t irredundant. 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]). [(G_t)_G is the normal core
of G_t in G (i.e., the largest normal subgroup of G contained in G_t).]
The reader may also consult my paper "Finite covers of groups by cosets
or subgroups" [Internat. J. Math. 17(2006), no.9, 1047-1064] for other
related results. These papers are available from my homepage. Here is a
link to my recent talk
"Study covers of groups via characters and number theory"
http://math.nju.edu.cn/~zwsun/CoverG.pdf It seems that my above conjecture is very challenging and its proof
might involve the representation theory of finite groups.
Zhi-Wei Sun (Nanjing University, P. R. China)
