Generalized Frobenius groups V. D. Mazurov Kazan June 2019 Thanks • This is a joint work with D. V.Lytkina and A. Kh. Zhurtov • The work is supported by RFBR grant No. 19-01-00507 2/21 Intro Definition Let Ω be a finite nonempty set and G be a transitive group on Ω with the following property: g Gα = fg 2 G j α = αg is nontrivial for some α 2 Ω and g g Gα,β = fg 2 G j α = α; β = βg is trivial for α; β 2 Ω, α 6= β. 3/21 Intro • G. Frobenius (1901) proved that in described situation the set G = f1g [ fg 2 G j αg 6= α for every α 2 Ωg forms a normal subgroup of G. • Since then such groups became known as Frobenius groups. G. Frobenius. Uber¨ auflosbare¨ Gruppen. IV. Berl. Ber. (1901), 1216–1230. 4/21 Intro Remark • Every Frobenius group as an abstract group is isomorphic to the semidirect product G = FH, where F is a proper nontrivial normal subgroup of G (the kernel of G), and a complement H which acts fixed-point freely on F , that is f h = h−1fh 6= f for every 1 6= f 2 F and every 1 6= h 2 H. 5/21 • F is nilpotent [G. Thompson, 1960]. J. G. Thompson. Normal p-complements for finite groups. Math. Z. 72 (1960), 332–354. Intro • Hence H acts on F # = F n f1g by conjugation in G semiregularly, that is, every orbit of H is of length jHj, so jF n f1gj = mjHj, where m is the number of orbits. In particular, jHj = 1 + mjF j is coprime to jF j. 6/21 Intro • Hence H acts on F # = F n f1g by conjugation in G semiregularly, that is, every orbit of H is of length jHj, so jF n f1gj = mjHj, where m is the number of orbits. In particular, jHj = 1 + mjF j is coprime to jF j. • F is nilpotent [G. Thompson, 1960]. J. G. Thompson. Normal p-complements for finite groups. Math. Z. 72 (1960), 332–354. 6/21 Intro • If H is insoluble then it contains a normal subgroup N ' SL2(5) and H=N is soluble [H. Zassenhaus, 1936; J. Wolf (1972): a corrected proof, in connection with classification of full connected Riemann manifolds of constant positive curvative; V.Mazurov (2003): short elementary proof]. H. Zassenhaus. Uber¨ endliche Fastkorper.¨ Abhandl. Math. Seminar Hamburg. Univ. 11 (1936), 187–220. J. A. Wolf. Spaces of constant curvative. Berkley, 1972. V.D. Mazurov. A new proof of Zassenhaus theorem on finite groups of fixed-point-free automorphisms. J. Algebra, 263, No. 1 (2003), 1–7. 7/21 n Z(SLn(q)) = fαE j α 2 GF (q); α = 1g; Ln(q) = P SLn(q) = SLn(q)=Z(SLn(q)) is a simple nonabelian group, except for the cases (n; q) = f(2; 2); (2; 3)g; L2(5) = SL2(5)=Z(SL2(5)) ' A5, the simple group of order 60. Explanation for designation SL2(5) SLn(q) = fa 2 Matn×n(GF (q)) j det a = 1g; 8/21 Ln(q) = P SLn(q) = SLn(q)=Z(SLn(q)) is a simple nonabelian group, except for the cases (n; q) = f(2; 2); (2; 3)g; L2(5) = SL2(5)=Z(SL2(5)) ' A5, the simple group of order 60. Explanation for designation SL2(5) SLn(q) = fa 2 Matn×n(GF (q)) j det a = 1g; n Z(SLn(q)) = fαE j α 2 GF (q); α = 1g; 8/21 L2(5) = SL2(5)=Z(SL2(5)) ' A5, the simple group of order 60. Explanation for designation SL2(5) SLn(q) = fa 2 Matn×n(GF (q)) j det a = 1g; n Z(SLn(q)) = fαE j α 2 GF (q); α = 1g; Ln(q) = P SLn(q) = SLn(q)=Z(SLn(q)) is a simple nonabelian group, except for the cases (n; q) = f(2; 2); (2; 3)g; 8/21 Explanation for designation SL2(5) SLn(q) = fa 2 Matn×n(GF (q)) j det a = 1g; n Z(SLn(q)) = fαE j α 2 GF (q); α = 1g; Ln(q) = P SLn(q) = SLn(q)=Z(SLn(q)) is a simple nonabelian group, except for the cases (n; q) = f(2; 2); (2; 3)g; L2(5) = SL2(5)=Z(SL2(5)) ' A5, the simple group of order 60. 8/21 Camina pair Definition Let F be a proper nontrivial normal subgroup of a group G. Then (G; F ) is a Camina pair, if g is conjugated to every element of the coset F g for every g 2 G n F . • The study of Camina pairs was initiated by the paper of A. Camina [1978]. • Today, a bibliography on Camina pairs has more than sixty entries (see the survey of M. Lewis [2018]). A. R. Camina. Some conditions which almost characterize Frobenius groups. Israel J. Math., 31 (1978), 153–160. M. L. Lewis. Camina groups, Camina pairs, and generalization, in: N. S. N. Sastry and M. K. Yadav (eds.), Group theory and computation. Springer Nature Singapore Pte Ltd (2018), 141–174. 9/21 Camina pairs Proposition 1 (1) If G is a finite Frobenius group with kernel F , then (G; F ) is a Camina pair. (2) [Arad, Mann, Muzychuk, Pech, 2009] If (G; F ) is a finite Camina pair, G is insoluble and G is not a Frobenius group, then F is a p-group for a prime p and every insoluble composition factor of G 1 is isomorphic to L2(q) for some q. More exactly, (G=Op(G)) is isomorphic to one of the following groups: l l 1 SL2(p ), where l 2 N and p > 3; 2 SL2(5), and p = 3; 3 SL2(13), and p = 3; 4 SL2(5), and p > 7. 1 Here Op(X) denotes the largest normal p-subgroup of X, and X is the soluble coradical of X, the intersection of all normal subgroups N of X, such that X=N is soluble. Z. Arad, A. Mann, M. Muzychuk, C. Pech. On non-solvable Camina pairs. J. Algebra, 322, No. 7 (2009), 2286–2298. 10/21 Questions 1 Do there exist Camina pairs of types (2) and (3)? 2 Do there exist Camina pairs that satisfy (1) with p 6= 5? 3 For which p are there Camina pairs that satisfy (4)? Camina pairs 1 (G=Op(G)) is isomorphic to one of the following groups: l l (1) SL2(p ), where l 2 N and p > 3; (2) SL2(5), and p = 3; (3) SL2(13), and p = 3; (4) SL2(5), and p > 7. Notice that there exists an example of Camina pair of type (1), where jGj = 26 · 3 · 5 and pl = 5, three examples of type (4), where p = 11; 29; 59. 11/21 2 Do there exist Camina pairs that satisfy (1) with p 6= 5? 3 For which p are there Camina pairs that satisfy (4)? Camina pairs 1 (G=Op(G)) is isomorphic to one of the following groups: l l (1) SL2(p ), where l 2 N and p > 3; (2) SL2(5), and p = 3; (3) SL2(13), and p = 3; (4) SL2(5), and p > 7. Notice that there exists an example of Camina pair of type (1), where jGj = 26 · 3 · 5 and pl = 5, three examples of type (4), where p = 11; 29; 59. Questions 1 Do there exist Camina pairs of types (2) and (3)? 11/21 3 For which p are there Camina pairs that satisfy (4)? Camina pairs 1 (G=Op(G)) is isomorphic to one of the following groups: l l (1) SL2(p ), where l 2 N and p > 3; (2) SL2(5), and p = 3; (3) SL2(13), and p = 3; (4) SL2(5), and p > 7. Notice that there exists an example of Camina pair of type (1), where jGj = 26 · 3 · 5 and pl = 5, three examples of type (4), where p = 11; 29; 59. Questions 1 Do there exist Camina pairs of types (2) and (3)? 2 Do there exist Camina pairs that satisfy (1) with p 6= 5? 11/21 Camina pairs 1 (G=Op(G)) is isomorphic to one of the following groups: l l (1) SL2(p ), where l 2 N and p > 3; (2) SL2(5), and p = 3; (3) SL2(13), and p = 3; (4) SL2(5), and p > 7. Notice that there exists an example of Camina pair of type (1), where jGj = 26 · 3 · 5 and pl = 5, three examples of type (4), where p = 11; 29; 59. Questions 1 Do there exist Camina pairs of types (2) and (3)? 2 Do there exist Camina pairs that satisfy (1) with p 6= 5? 3 For which p are there Camina pairs that satisfy (4)? 11/21 • If G is such a group, then, for every p-element F g of the quotient group G=F (p is a prime), every element in F g is a p-element. • Notice that the class of Camina pairs is strictly embedded into the class EO. Example l G = V : SL2(p ), p is an odd prime, l 2 N, V is the natural l l 2-dimensional module for SL2(p ) over a field of order p . Camina pairs • If (G; F ) is a Camina pair, then, in particular, every coset F g, g 2 G n F consists of elements whose orders are equal, and we come to the following Problem Describe the class EO (from “equal orders”) of groups G having a proper nontrivial normal subgroup F , such that for every coset F g, where g 2 G n F , the order of every element in F g is equal to the order of g.