Structural criteria in factorised groups via conjugacy class sizes - - PowerPoint PPT Presentation
Structural criteria in factorised groups via conjugacy class sizes V ctor Manuel Ortiz-Sotomayor Universitat Polit` ecnica de Val` encia, Spain G ROUPS S T A NDREWS 2017 Birmingham, August 2017 Joint work with: Mar a Jos e
V´ ıctor Manuel Ortiz-Sotomayor
Universitat Polit` ecnica de Val` encia, Spain
GROUPS ST ANDREWS 2017
Birmingham, August 2017 Joint work with: ⊲ Mar´ ıa Jos´ e Felipe ⊲ Ana Mart´ ınez-Pastor
Introduction Square-free class sizes Prime power class sizes
All groups considered will be finite ⊲ Factorised groups: G = AB, where A and B are subgroups of a group G
Introduction Square-free class sizes Prime power class sizes
All groups considered will be finite ⊲ Factorised groups: G = AB, where A and B are subgroups of a group G Questions.
Introduction Square-free class sizes Prime power class sizes
All groups considered will be finite ⊲ Factorised groups: G = AB, where A and B are subgroups of a group G Questions.
Theorem (Fitting, 1938). If G = AB is the product of the nilpotent normal subgroups A and B, then G is nilpotent.
Introduction Square-free class sizes Prime power class sizes
All groups considered will be finite ⊲ Factorised groups: G = AB, where A and B are subgroups of a group G Questions.
Theorem (Fitting, 1938). If G = AB is the product of the nilpotent normal subgroups A and B, then G is nilpotent. Theorem (Kegel–Wielandt, 1962, 1958). If G = AB is the product of the nilpotent subgroups A and B, then G is soluble.
Introduction Square-free class sizes Prime power class sizes
⊲ Conjugacy class of x ∈ G: we denote xG := {xg : g ∈ G}, and its size is
Introduction Square-free class sizes Prime power class sizes
⊲ Conjugacy class of x ∈ G: we denote xG := {xg : g ∈ G}, and its size is
⇓ G = AB and to impose arithmetical conditions on
where x ∈ A ∪ B.
Introduction Square-free class sizes Prime power class sizes
⊲ Conjugacy class of x ∈ G: we denote xG := {xg : g ∈ G}, and its size is
⇓ G = AB and to impose arithmetical conditions on
where x ∈ A ∪ B. Arithmetical conditions.
Introduction Square-free class sizes Prime power class sizes
⊲ Conjugacy class of x ∈ G: we denote xG := {xg : g ∈ G}, and its size is
⇓ G = AB and to impose arithmetical conditions on
where x ∈ A ∪ B. Arithmetical conditions.
Some notes.
does not divide
.
Introduction Square-free class sizes Prime power class sizes
⊲ Conjugacy class of x ∈ G: we denote xG := {xg : g ∈ G}, and its size is
⇓ G = AB and to impose arithmetical conditions on
where x ∈ A ∪ B. Arithmetical conditions.
Some notes.
does not divide
.
Introduction Square-free class sizes Prime power class sizes
⊲ Conjugacy class of x ∈ G: we denote xG := {xg : g ∈ G}, and its size is
⇓ G = AB and to impose arithmetical conditions on
where x ∈ A ∪ B. Arithmetical conditions.
Some notes.
does not divide
.
If g ∈ G = AB, then gG = (ab)G = aGbG, and
is not related with
and
.
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB, where A and B are permutable subgroups of G. Suppose that
is square-free for every x ∈ A ∪ B. Then G is supersoluble.
Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups, J. Pure and Applied Algebra, 196: 111-117, 2005.
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB, where A and B are permutable subgroups of G. Suppose that
is square-free for every x ∈ A ∪ B. Then G is supersoluble.
Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups, J. Pure and Applied Algebra, 196: 111-117, 2005.
Remark. The factors A and B inherit the class size hypotheses, i.e.,
divides
, ∀ a ∈ A.
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB, where A and B are permutable subgroups of G. Suppose that
is square-free for every x ∈ A ∪ B. Then G is supersoluble.
Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups, J. Pure and Applied Algebra, 196: 111-117, 2005.
Remark. The factors A and B inherit the class size hypotheses, i.e.,
divides
, ∀ a ∈ A. Question. To what extent can the permutability hypothesis on the factors be weakened?
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB, where A and B are permutable subgroups of G. Suppose that
is square-free for every x ∈ A ∪ B. Then G is supersoluble.
Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups, J. Pure and Applied Algebra, 196: 111-117, 2005.
Corollary. Suppose that
is square-free for every x ∈ G. Then G is supersoluble (and both G/ F(G) and G′ are cyclic with square-free orders).
Chillag, D., and Herzog, M., On the length of the conjugacy classes of finite groups, J. Algebra, 131: 110-125, 1990. Cossey, J., and Wang, Y., Remarks on the length of conjugacy classes of finite groups, Comm. Algebra, 27: 4347-4353, 1999.
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB, where A and B are permutable subgroups of G. Suppose that
is square-free for every x ∈ A ∪ B. Then G is supersoluble.
Liu, X., Wang, Y., and Wei, H., Notes on the length of conjugacy classes of finite groups, J. Pure and Applied Algebra, 196: 111-117, 2005.
Corollary. Suppose that
is square-free for every x ∈ G. Then G is supersoluble (and both G/ F(G) and G′ are cyclic with square-free orders).
Chillag, D., and Herzog, M., On the length of the conjugacy classes of finite groups, J. Algebra, 131: 110-125, 1990. Cossey, J., and Wang, Y., Remarks on the length of conjugacy classes of finite groups, Comm. Algebra, 27: 4347-4353, 1999.
Question. Can we obtain further information for factorised groups?
Introduction Square-free class sizes Prime power class sizes
G = AB : A, B supersoluble + A, B G G supersoluble Question. G = AB : A, B supersoluble + permutability properties + additional conditions. Then G is supersoluble.
Introduction Square-free class sizes Prime power class sizes
G = AB : A, B supersoluble + A, B G G supersoluble Question. G = AB : A, B supersoluble + permutability properties + additional conditions. Then G is supersoluble. Theorem. Let G = AB with A and B supersoluble, and assume that G′ is nilpotent. If A and B are mutually permutable, then G is supersoluble.
Asaad, M., and Shaalan, A., On the supersolvability of finite groups, Arch. Math. (Basel), 53: 318-326, 1989.
Definition. A and B are mutually permutable if A permutes with every subgroup of B and B permutes with every subgroup of A.
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB such that A and B are mutually permutable. Assume
is square-free for every element x ∈ A ∪ B. Then we have: ⊲ G is supersoluble. ⊲ G′ is abelian with elementary abelian Sylow subgroups. ⊲ G/ F(G) has elementary abelian Sylow subgroups. ⊲ The structure of the Sylow p-subgroups of G/ F(G) is either Cp or Cp × Cp, ∀ p.
Felipe, M. J., Mart´ ınez-Pastor, A., and Ortiz-Sotomayor, V. M., Square-free class sizes in products of groups, submitted.
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB such that A and B are mutually permutable. Assume
is square-free for every element x ∈ A ∪ B. Then we have: ♮ G is supersoluble. ⊲ G′ is abelian with elementary abelian Sylow subgroups. ♮ G/ F(G) has elementary abelian Sylow subgroups. ⊲ The structure of the Sylow p-subgroups of G/ F(G) is either Cp or Cp × Cp, ∀ p.
Felipe, M. J., Mart´ ınez-Pastor, A., and Ortiz-Sotomayor, V. M., Square-free class sizes in products of groups, submitted.
Remark. ♮ remains true if we assume p2 ∤
for all p-regular elements x ∈ A ∪ B, for every p. Corollary. Let G = AB such that A and B are mutually permutable. If p2 ∤
for all p-regular elements x ∈ A ∪ B, and for every prime p, then G is supersoluble.
Ballester-Bolinches, A., Cossey, J., and Li, Y., Mutually permutable products and conjugacy classes,
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB such that A and B are mutually permutable. Assume
is square-free for every element x ∈ A ∪ B. Then we have: † G is supersoluble. † G′ is abelian with elementary abelian Sylow subgroups. † G/ F(G) has elementary abelian Sylow subgroups. ⊲ The structure of the Sylow p-subgroups of G/ F(G) is either Cp or Cp × Cp, ∀ p.
Felipe, M. J., Mart´ ınez-Pastor, A., and Ortiz-Sotomayor, V. M., Square-free class sizes in products of groups, submitted.
Remark. † remains true if we consider only prime power order elements in A ∪ B. Example. Let {p1, . . . , pn} be a set of pairwise distinct odd primes, and let G = A × B = D2p1 × (D2p2 × · · · × D2pn). Every prime power order element x ∈ A ∪ B has square-free
, but |G/ F(G)| = 2n
Introduction Square-free class sizes Prime power class sizes
Theorem. If P = AB is a p-group such that p2 ∤
for every x ∈ A ∪ B, then: ⊲ P′ Φ(P) Z(P). ⊲ P′ is elementary abelian. ⊲ |P′| ≤ p2.
F, M-P , O-S, Square-free class sizes in products of groups, submitted.
Introduction Square-free class sizes Prime power class sizes
Theorem. If P = AB is a p-group such that p2 ∤
for every x ∈ A ∪ B, then: ⊲ P′ Φ(P) Z(P). ⊲ P′ is elementary abelian. ⊲ |P′| ≤ p2.
F, M-P , O-S, Square-free class sizes in products of groups, submitted.
Theorem. If P is a p-group, then p2 ∤
for every x ∈ P if and only if |P′| ≤ p.
Knoche, H. G., ¨ Uber den Frobeniusschen Klassenbegriff in nilpotenten Gruppen, Math. Z., 55: 71-83, 1951.
Introduction Square-free class sizes Prime power class sizes
Theorem. If P = AB is a p-group such that p2 ∤
for every x ∈ A ∪ B, then: ⊲ P′ Φ(P) Z(P). ⊲ P′ is elementary abelian. ⊲ |P′| ≤ p2.
F, M-P , O-S, Square-free class sizes in products of groups, submitted.
Theorem. If P is a p-group, then p2 ∤
for every x ∈ P if and only if |P′| ≤ p.
Knoche, H. G., ¨ Uber den Frobeniusschen Klassenbegriff in nilpotenten Gruppen, Math. Z., 55: 71-83, 1951.
Example. The converse is not true (in contrast to Knoche’s theorem): P = AB = [C4]Q8 = x, i, j | x4 = i4 = j4 = 1, i2j2 = 1, jij = i, xi = x−1, [x, j] = 1. It follows that P′ = Φ(P) = Z(P) = C2 × C2, but CP(i) = x2 × i, so
= 4.
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB such that A and B are mutually permutable. Assume (p − 1, |G|) = 1. If p2 ∤
for all p-regular elements x ∈ A ∪ B of prime power order, then: ⊲ G is soluble. ⊲ G is p-nilpotent. ⊲ G/ Op(G) has elementary abelian Sylow p-subgroups.
F, M-P , O-S, Square-free class sizes in products of groups, submitted.
Introduction Square-free class sizes Prime power class sizes
Theorem. Let G = AB such that A and B are mutually permutable. Assume (p − 1, |G|) = 1. If p2 ∤
for all p-regular elements x ∈ A ∪ B of prime power order, then: ⊲ G is soluble. ⊲ G is p-nilpotent. ⊲ G/ Op(G) has elementary abelian Sylow p-subgroups.
F, M-P , O-S, Square-free class sizes in products of groups, submitted.
Example. Let G = [C5 × C5](Sym(3) × C2) = SmallGroup(300, 25) - GAP. Then G = AB, where A = D10 × D10 and B = [C5 × C5]C3 are mutually permutable. It holds that 4 ∤
for every 2-regular element x ∈ A ∪ B of prime power order. Nevertheless, ∃ a ∈ A of prime power order and 2-regular such that 4 divides
.
Introduction Square-free class sizes Prime power class sizes
Theorem (Baer, 1953).
is a prime power ∀ x ∈ G of prime power order ⇐ ⇒ G = G1 × · · · × Gr such that
Introduction Square-free class sizes Prime power class sizes
Theorem (Baer, 1953).
is a prime power ∀ x ∈ G of prime power order ⇐ ⇒ G = G1 × · · · × Gr such that
Remark. In 1998, Camina and Camina made a shorter proof by analysing the so-called p-Baer groups. Definition. ⊲ G is a p-Baer group if every p-element x ∈ G has
a prime power. ⊲ G is a Baer group ⇐ ⇒ G is a p-Baer group for every p ∈ π(G).
Introduction Square-free class sizes Prime power class sizes
Main idea ⇓ G = AB and p ∈ π(G) = ⇒ ∃ P ∈ Sylp (G) such that P = (P ∩ A)(P ∩ B) P ∩ A ∈ Sylp (A) P ∩ B ∈ Sylp (B)
Introduction Square-free class sizes Prime power class sizes
Main idea ⇓ G = AB and p ∈ π(G) = ⇒ ∃ P ∈ Sylp (G) such that P = (P ∩ A)(P ∩ B) P ∩ A ∈ Sylp (A) P ∩ B ∈ Sylp (B) Some key facts. ⊲ If P = (P ∩ A)(P ∩ B) and P ∩ A F(G), then Op(G) = (Op(G) ∩ A)(Op(G) ∩ B). ⊲ If x ∈ G has prime power
, then x ∈ F2(G) (due to Camina and Camina, 1998).
Introduction Square-free class sizes Prime power class sizes
Theorem. If G = AB is a p-Baer factorisation, and P ∈ Sylp (G), then: ⊲ P F(G) G and P Op′(G) G. So G is p-soluble of p-length 1.
Introduction Square-free class sizes Prime power class sizes
Theorem. If G = AB is a p-Baer factorisation, and P ∈ Sylp (G), then: ⊲ P F(G) G and P Op′(G) G. So G is p-soluble of p-length 1. ⊲ ∃! q :
is a q-power for all p-elements x ∈ A, and ∃! r :
is an r-power for all p-elements y ∈ B.
Introduction Square-free class sizes Prime power class sizes
Theorem. If G = AB is a p-Baer factorisation, and P ∈ Sylp (G), then: ⊲ P F(G) G and P Op′(G) G. So G is p-soluble of p-length 1. ⊲ ∃! q :
is a q-power for all p-elements x ∈ A, and ∃! r :
is an r-power for all p-elements y ∈ B. ⊲ P Oq(G) Or(G) is normal in G.
Introduction Square-free class sizes Prime power class sizes
Theorem. If G = AB is a p-Baer factorisation, and P ∈ Sylp (G), then: ⊲ P F(G) G and P Op′(G) G. So G is p-soluble of p-length 1. ⊲ ∃! q :
is a q-power for all p-elements x ∈ A, and ∃! r :
is an r-power for all p-elements y ∈ B. ⊲ P Oq(G) Or(G) is normal in G. ⊲ If q = r = p, then G is p-decomposable, that is, G = Op(G) × Op′(G).
Introduction Square-free class sizes Prime power class sizes
Theorem. If G = AB is a p-Baer factorisation, and P ∈ Sylp (G), then: ⊲ P F(G) G and P Op′(G) G. So G is p-soluble of p-length 1. ⊲ ∃! q :
is a q-power for all p-elements x ∈ A, and ∃! r :
is an r-power for all p-elements y ∈ B. ⊲ P Oq(G) Or(G) is normal in G. ⊲ If q = r = p, then G is p-decomposable, that is, G = Op(G) × Op′(G). ⊲ If p / ∈ {q, r}, then P is abelian.
Introduction Square-free class sizes Prime power class sizes
Theorem. If G = AB is a p-Baer factorisation, and P ∈ Sylp (G), then: ⊲ P F(G) G and P Op′(G) G. So G is p-soluble of p-length 1. ⊲ ∃! q :
is a q-power for all p-elements x ∈ A, and ∃! r :
is an r-power for all p-elements y ∈ B. ⊲ P Oq(G) Or(G) is normal in G. ⊲ If q = r = p, then G is p-decomposable, that is, G = Op(G) × Op′(G). ⊲ If p / ∈ {q, r}, then P is abelian. ⊲ G/ CG(Op(G)) is p-decomposable.
Introduction Square-free class sizes Prime power class sizes
Theorem. If G = AB is a p-Baer factorisation, and P ∈ Sylp (G), then: ⊲ P F(G) G and P Op′(G) G. So G is p-soluble of p-length 1. ⊲ ∃! q :
is a q-power for all p-elements x ∈ A, and ∃! r :
is an r-power for all p-elements y ∈ B. ⊲ P Oq(G) Or(G) is normal in G. ⊲ If q = r = p, then G is p-decomposable, that is, G = Op(G) × Op′(G). ⊲ If p / ∈ {q, r}, then P is abelian. ⊲ G/ CG(Op(G)) is p-decomposable. ⊲ The Sylow p-subgroup of G/ F(G) is abelian.
Introduction Square-free class sizes Prime power class sizes
Theorem. If G = AB is a p-Baer factorisation, and P ∈ Sylp (G), then: ⊲ P F(G) G and P Op′(G) G. So G is p-soluble of p-length 1. ⊲ ∃! q :
is a q-power for all p-elements x ∈ A, and ∃! r :
is an r-power for all p-elements y ∈ B. ⊲ P Oq(G) Or(G) is normal in G. ⊲ If q = r = p, then G is p-decomposable, that is, G = Op(G) × Op′(G). ⊲ If p / ∈ {q, r}, then P is abelian. ⊲ G/ CG(Op(G)) is p-decomposable. ⊲ The Sylow p-subgroup of G/ F(G) is abelian. ⊲ P is abelian if and only if Op(G) so is.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Introduction Square-free class sizes Prime power class sizes
Theorem. If G = AB is a p-Baer factorisation, and P ∈ Sylp (G), then: † P F(G) G and P Op′(G) G. So G is p-soluble of p-length 1. † ∃! q :
is a q-power for all p-elements x ∈ A, and ∃! r :
is an r-power for all p-elements y ∈ B. † P Oq(G) Or(G) is normal in G. † If q = r = p, then G is p-decomposable, that is, G = Op(G) × Op′(G). † If p / ∈ {q, r}, then P is abelian. ⊲ G/ CG(Op(G)) is p-decomposable. ⊲ The Sylow p-subgroup of G/ F(G) is abelian. ⊲ P is abelian if and only if Op(G) so is.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Corollary. If G = A = B is a p-Baer group, then it follows the above statements † for r = q.
Camina, A. R., and Camina, R. D., Implications of conjugacy class size, J. Group Theory, 1(3): 257–269, 1998.
Introduction Square-free class sizes Prime power class sizes
Proposition. If G = AB is a Baer factorisation, then: ⊲ G/ F(G) is abelian. ⊲ Let a ∈ A of prime power order. If
is a q-number, then
is also a q-number.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Introduction Square-free class sizes Prime power class sizes
Proposition. If G = AB is a Baer factorisation, then: ⊲ G/ F(G) is abelian. ⊲ Let a ∈ A of prime power order. If
is a q-number, then
is also a q-number.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Example. Let G = [C2 × C2 × C2]([C7]C3) such that C7 permutes transitively the involutions. For certain A ∈ Hall{2,3} (G), B ∈ Syl7 (G), we get that G = AB is a 2-Baer factorisation. Nevertheless, there is a 2-element a ∈ A such that
= 3 and
= 7.
Introduction Square-free class sizes Prime power class sizes
Proposition. If G = AB is a Baer factorisation, then: ⊲ G/ F(G) is abelian. ⊲ Let a ∈ A of prime power order. If
is a q-number, then
is also a q-number.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Example. Let G = [C2 × C2 × C2]([C7]C3) such that C7 permutes transitively the involutions. For certain A ∈ Hall{2,3} (G), B ∈ Syl7 (G), we get that G = AB is a 2-Baer factorisation. Nevertheless, there is a 2-element a ∈ A such that
= 3 and
= 7. Corollary. If G = AB is a Baer factorisation, then A and B are Baer groups.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Introduction Square-free class sizes Prime power class sizes
Proposition. If G = AB is a Baer factorisation, then: ⊲ G/ F(G) is abelian. ⊲ Let a ∈ A of prime power order. If
is a q-number, then
is also a q-number.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Example. Let G = [C2 × C2 × C2]([C7]C3) such that C7 permutes transitively the involutions. For certain A ∈ Hall{2,3} (G), B ∈ Syl7 (G), we get that G = AB is a 2-Baer factorisation. Nevertheless, there is a 2-element a ∈ A such that
= 3 and
= 7. Corollary. If G = AB is a Baer factorisation, then A and B are Baer groups.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Open question. If G = AB is a p-Baer factorisation, then A and B are p-Baer groups.
Introduction Square-free class sizes Prime power class sizes
Example. Let G = A × B = (C3 × [C7]C2 × [C11]C5) × (C5 × [C7]C3 × [C11]C2). This is a Baer factorisation, but there are no pairwise coprime proper direct factors of G.
Introduction Square-free class sizes Prime power class sizes
Example. Let G = A × B = (C3 × [C7]C2 × [C11]C5) × (C5 × [C7]C3 × [C11]C2). This is a Baer factorisation, but there are no pairwise coprime proper direct factors of G. We have attained an arithmetical characterisation for Baer factorisations: Theorem. G = AB is a Baer factorisation ⇐ ⇒ |G : CG(Ap)| and |G : CG(Bp)| are prime powers, for Ap ∈ Sylp (A) and Bp ∈ Sylp (B), and for all p.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Introduction Square-free class sizes Prime power class sizes
Example. Let G = A × B = (C3 × [C7]C2 × [C11]C5) × (C5 × [C7]C3 × [C11]C2). This is a Baer factorisation, but there are no pairwise coprime proper direct factors of G. We have attained an arithmetical characterisation for Baer factorisations: Theorem. G = AB is a Baer factorisation ⇐ ⇒ |G : CG(Ap)| and |G : CG(Bp)| are prime powers, for Ap ∈ Sylp (A) and Bp ∈ Sylp (B), and for all p.
F, M-P , O-S, Prime power indices in factorised groups, submitted.
Example. G = [C2 × C2 × C2]([C7]C3) = [A]B such that C7 permutes transitively the involutions. Then G = AB is a 2-Baer factorisation, but |G : CG(Ap)| = 21.
V´ ıctor Manuel Ortiz-Sotomayor
Universitat Polit` ecnica de Val` encia, Spain
GROUPS ST ANDREWS 2017
Birmingham, August 2017 Joint work with: ⊲ Mar´ ıa Jos´ e Felipe ⊲ Ana Mart´ ınez-Pastor