Groups with subnormal normalizers of subnormal subgroups. (Q2907017)

Results are obtained for finite soluble groups in which all subnormal subgroups have subnormal normalisers, so called NSS-groups.NEWLINENEWLINE The major result is Theorem 10: If \(G\) is an NSS-group, then (a) \(G\) has \(p\)-length 1 for all primes \(p\); (b) \(G/F_2\) is nilpotent of squarefree exponent, where \(F_2\) is the largest normal subgroup of \(G\) of Fitting length at most 2, so that \(G\) has Fitting length at most 3; (c) \(Q'\) and \(Q/Z(Q)\) are of exponent 2, where \(Q=G^{\mathcal N}F(G)/F(G)\).NEWLINENEWLINE More specific results are provided for certain subclasses of the class of NSS-groups, for example: the normaliser of each subnormal subgroup of the soluble group \(G\) contains \(M=G^{\mathcal N}\) if and only if every Sylow \(p\)-subgroup of \(G\) satisfies the following conditions: (i) there is a supplement \(C_S\) of \(S\cap M\) in \(S\) such that \([C_S,S\cap M]=1\) and (ii) if \(M'\neq 1\) and \(p=2\), then \(C_S\) is elementary Abelian.