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.

SUBNORMALITY IN THE JOIN OF TWO SUBGROUPS.

DARDANO, ULDERICO
2004

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11588/103726
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