Abstract
According to Hall, a subgroup H of a group G is said to be pronormal if H and \(H^g\) are conjugate in \(\langle H,H^g\rangle \) for every \(g\in G\). In this survey, we discuss the role of pronormality for some subgroups of finite groups: Hall subgroups, subgroups of odd index, submaximal \(\mathfrak X\)-subgroup, etc.
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Notes
Recall that a normal subset D of a group G is a set of odd transpositions (resp., 3-transpositions) if \(|t|=2\) and |st| either equals 2 or is an odd number (resp., \(|st|\in \{1,2,3\}\)) for every \(s,t\in D\).
Recall that if K is a subgroup of G, then an overgroup of K is a subgroup of G containing K.
Recall that G is said to be almost simple if its socle is a non-Abelian simple group. In other words, G is isomorphic to a subgroup of Aut(S) that contains Inn(S) for some non-Abelian simple group S.
Recall that every \(\langle \textsc {s}_n,\textsc {n}_0\rangle \)-closed class of groups is called a Fitting class.
First inclusion follows from the Sylow theorem and from the solvability of groups of prime power order. The second one is obvious.
Wielandt’s proof can be found in his lectures [71, 13.2].
Wielandt refers to a subgroup H of a group G as intravariant if its conjugacy class in G is invariant under the natural action of the group \({{\mathrm{Aut}}}(G)\) on the conjugacy classes of subgroups.
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The first author is supported by a NNSF Grant of China (Grant # 11771409) and Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences and by SB RAS Fundamental Research Program I.1.1 (project # 0314-2016-0001).
Appendix: Submaximal \(\pi \)-Subgroups of Minimal Simple Groups
Appendix: Submaximal \(\pi \)-Subgroups of Minimal Simple Groups
In Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 we use the following notation.
The conditions in the column ‘Cond.’ are necessary and sufficient for the existence and the \(\pi \)-submaximality of corresponding H. If a cell in this column is empty, then it means that the corresponding \(\pi \)-submaximal subgroup always exists.
In the column ‘H’ the structure of corresponding H is given.
The conditions in the column ‘is not \(\pi \)-max. if’ are necessary and sufficient for corresponding H to be not \(\pi \)-maximal in S. If either this column is skipped or a cell in this column is empty, then the corresponding subgroup is \(\pi \)-maximal.
A number n in the column ‘NCC’ is equal to the number of conjugacy classes of \(\pi \)-submaximal subgroups of S isomorphic to corresponding subgroup H, and if \(n>1\), then in the same column the action of \({{\mathrm{Aut}}}(S)\) on these classes is described.
The symbol ‘\(\checkmark \)’ in the column ‘Pro.’ means that the corresponding subgroup H is pronormal in S.
The symbol ‘\(\checkmark \)’ in the column ‘Intra.’ means that the corresponding subgroup H is intravariant in S. If a cell in this column is empty, then H is not intravariant.
1.1 The \(\pi \)-Submaximal Subgroups in \(S=L_2(q)\), Where \(q=2^p\), p is Prime, for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)
1.2 The \(\pi \)-Submaximal Subgroups in \(S=L_2(q)\), Where \(q=3^p\), p is Odd Prime, for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)
1.3 The \(\pi \)-Submaximal Subgroups in \(S=L_2(q)\), Where q is a Prime, \(q^2\equiv -1\pmod 5\), for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)
1.4 The Submaximal \(\pi \)-Subgroups in \(S=Sz(q)\), Where \(q=2^p\), p is Odd Prime, for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)
1.5 The \(\pi \)-Submaximal Subgroups of \(S=L_3(3)\), for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)
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Guo, W., Revin, D.O. Pronormality and Submaximal \(\mathfrak {X}\)-Subgroups on Finite Groups. Commun. Math. Stat. 6, 289–317 (2018). https://doi.org/10.1007/s40304-018-0154-9
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DOI: https://doi.org/10.1007/s40304-018-0154-9