SUBNORMALITY IN THE JOIN OF TWO SUBGROUPS.

In the framework of Group Theory, we consider the problem of proving (or disproving) the subnormality of a subgroup H (of a group G) which is contained and subnormal in two subgroups U and V that together generate the whole group G. It was known that the answer is yes when either G is finite or G=UV is the product of U and V. In the paper it is shown that the answer remains true even when the derived subgroup G' of G (possibly infinite) is nilpotent. On the other hand, we show that the answer is no if G is infinite even if G=UV (and G is locally soluble and U and V are locally nilpotent). In the affirmative case, we show that there is a polynomial function bounding the subnormality defect of H in G in terms of the defects in U and G (and the nilpotency class of $G'$). We also show that this holds even if we replace subnormality by ascendancy.