Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations
of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any
one of n cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms
and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic
distributions for the total winnings of the n players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For
all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that
for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic
expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be
numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations
even for very small n.
Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(p). Zhang  proved that if P is a Sylow p-subgroup of G, Ω1(P) ≦ Z(P) and NG(Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result.