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Journal of the London Mathematical Society Advance Access published online on September 24, 2007
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm050
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© 2007 London Mathematical Society

On subnormality criteria for subgroups in finite groups

Francesco Fumagalli

Dipartimento di Matematica ‘Ulisse Dini’
Università degli Studi di Firenze
viale Morgagni 67A
50134 Firenze
Italy

Let H be a subgroup of a finite group G and let Formula be the set of all elements g of G such that H is subnormal in <H, Hg>. A result of Wielandt states that H is subnormal in G if and only if Formula. In this paper, we let A be a subgroup of G contained in Formula and ask if this implies (and therefore is equivalent to) the subnormality of H in <H, A>. We show with an example that the answer is no, even for soluble groups with Sylow subgroups of nilpotency class at most 2. However, we prove that the two conditions are equivalent whenever A either is subnormal in G or has p-power index in G (for p any prime number).

fumagalli{at}math.unifi.it

2000 Mathematics Subject Classification 20D35.

This work was partially supported by the MURST researchprogram ‘Teoria dei gruppi e applicazioni’.

Received September 7, 2005; revised November 22, 2006;
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