\(\Pi\)-normally embedded subgroups of finite soluble groups (Q1915407)

All groups considered in this paper are finite and soluble. Let \(G\) be a group and \(\pi\) a set of primes. A subgroup \(H\) of \(G\) is said to be \(\pi\)-normally embedded in \(G\) if a Hall \(\pi\)-subgroup of \(H\) is a Hall \(\pi\)-subgroup of some normal subgroup of \(G\). Theorem 1: Let \(U\) and \(V\) be \(\pi\)-normally embedded subgroups of \(G\). Then \(U \cap V\) is \(\pi\)-normally embedded in \(G\) provided that one of the following conditions is satisfied: i) \(U\) permutes with \(V\); ii) there exists a Hall \(\pi\)-subgroup of \(G\) that reduces into \(U\) and \(V\); iii) \(U\) is a subnormal subgroup of \(G\); iv) \(U \cap V\) is a nilpotent normal subgroup of \(G\). Theorem 2: Assume that \(U\) and \(V\) are \(\pi\)-normally embedded subgroups of a group \(G\). Then \(\langle U,V\rangle\) is \(\pi\)-normally embedded in \(G\) provided that one of the following conditions is satisfied: i) \(U\) permutes with \(V\); ii) there exists a Hall system \(\Sigma\) of \(G\) which reduces into \(U\) and \(V\); iii) either \(U\) or \(V\) is a subnormal subgroup of \(\langle U, V\rangle\). In particular, the two theorems put together generalise an unpublished result by B. Fischer [\textit{K. Doerk} and \textit{T. Hawkes}, Finite Soluble Groups, Walter de Gruyter, Berlin-New York (1992; Zbl 0753.20001), Theorem 1.7.9] that was also proved by \textit{F. Lockett} [On the theory of Fitting classes of finite soluble groups, Ph. D. thesis, Univ. Warwick (1971)] and by \textit{T. Yen} [Proc. Am. Math. Soc. 34, 340-342 (1972; Zbl 0255.20012)].