On sets of generators of finite groups Agnieszka Stocka University of Białystok Groups and their action 22-26.06.2015 A. Stocka On sets of generators of finite groups 1 / 28 Notations and Definitions A. Stocka On sets of generators of finite groups 2 / 28 Notations and Definitions • All group considered are finite. A. Stocka On sets of generators of finite groups 2 / 28 Notations and Definitions • All group considered are finite. • By Φ(G) we denoted the Frattini subgroup of a group G that is the set of all nongenerators of G. A. Stocka On sets of generators of finite groups 2 / 28 Notations and Definitions • All group considered are finite. • By Φ(G) we denoted the Frattini subgroup of a group G that is the set of all nongenerators of G. Definition 1 A subset X of G is called: A. Stocka On sets of generators of finite groups 2 / 28 Notations and Definitions • All group considered are finite. • By Φ(G) we denoted the Frattini subgroup of a group G that is the set of all nongenerators of G. Definition 1 A subset X of G is called: • g-independent if hY, Φ(G)i = 6 hX, Φ(G)i for every proper subset Y of X; A. Stocka On sets of generators of finite groups 2 / 28 Notations and Definitions • All group considered are finite. • By Φ(G) we denoted the Frattini subgroup of a group G that is the set of all nongenerators of G. Definition 1 A subset X of G is called: • g-independent if hY, Φ(G)i = 6 hX, Φ(G)i for every proper subset Y of X; • a generating set of G if hXi = G (h, Φ(G)i = G); A. Stocka On sets of generators of finite groups 2 / 28 Notations and Definitions • All group considered are finite. • By Φ(G) we denoted the Frattini subgroup of a group G that is the set of all nongenerators of G. Definition 1 A subset X of G is called: • g-independent if hY, Φ(G)i = 6 hX, Φ(G)i for every proper subset Y of X; • a generating set of G if hXi = G (h, Φ(G)i = G); • a g-base of G, if X is a g-independent generating set of G. A. Stocka On sets of generators of finite groups 2 / 28 Definitions Proposition 2 Let G be any group, X ⊂ G a subset and X the natural image of X in G/Φ(G). Then: A. Stocka On sets of generators of finite groups 3 / 28 Definitions Proposition 2 Let G be any group, X ⊂ G a subset and X the natural image of X in G/Φ(G). Then: • X is g-independent in G if and only if X is g-independent in G/Φ(G); A. Stocka On sets of generators of finite groups 3 / 28 Definitions Proposition 2 Let G be any group, X ⊂ G a subset and X the natural image of X in G/Φ(G). Then: • X is g-independent in G if and only if X is g-independent in G/Φ(G); • X is a generating set of G if and only if X is a generating set of G/Φ(G); A. Stocka On sets of generators of finite groups 3 / 28 Definitions Proposition 2 Let G be any group, X ⊂ G a subset and X the natural image of X in G/Φ(G). Then: • X is g-independent in G if and only if X is g-independent in G/Φ(G); • X is a generating set of G if and only if X is a generating set of G/Φ(G); • X is a g-base of G if and only if X is a g-base of G/Φ(G). A. Stocka On sets of generators of finite groups 3 / 28 Definitions Definition 3 We say that a group G • has property B (is a B-group) if every two g-bases of G have the same size; A. Stocka On sets of generators of finite groups 4 / 28 Definitions Definition 3 We say that a group G • has property B (is a B-group) if every two g-bases of G have the same size; • has the embedding property if every g-independent subset of G is contained in a g-base of G; A. Stocka On sets of generators of finite groups 4 / 28 Definitions Definition 3 We say that a group G • has property B (is a B-group) if every two g-bases of G have the same size; • has the embedding property if every g-independent subset of G is contained in a g-base of G; • is matroid if G has property B and the embedding property. A. Stocka On sets of generators of finite groups 4 / 28 Definitions Definition 3 We say that a group G • has property B (is a B-group) if every two g-bases of G have the same size; • has the embedding property if every g-independent subset of G is contained in a g-base of G; • is matroid if G has property B and the embedding property. Proposition 4 A group G has property B, the embedding property, is matroid if and only if G/Φ(G) has property B, the embedding property, is matroid, respectively. A. Stocka On sets of generators of finite groups 4 / 28 Examples Theorem 5 (Burnside Basis Theorem,1912) Let G be a p-group and |G/Φ(G)| = pd . Then G/Φ(G) is an elementary abelian p-group (a vector space over Fp ) and every g-base of G has d generators. A. Stocka On sets of generators of finite groups 5 / 28 Examples Let • p, q be primes such that q m |pn − 1; • V be the additive group of Fpn isomorphic to n direct product of Cp . • H be a subgroup of the multiplikative group of the field Fpn of order q m ; • ϕ : H −→ AutV be a mapping such that hϕ is the automorphism of V and xϕ : v 7−→ vx, for every v ∈ V. Group isomorphic to a semidirect product G = V oϕ H will be called a scalar extension. The cyclic group of order q m acts on the left hand side by multiplication in the field Fpn and we can see that ϕ induces upon V the structure of an Fp H -module. A. Stocka On sets of generators of finite groups 6 / 28 References • P. Apisa, B. Klopsch, A generalization of the Burnside basis theorem, J. Algebra 400 (2014), 8–16. • J. McDougall-Bagnall, M. Quick, Groups with the basis property, J. Algebra 346 (2011), 332–339. • R. Scapellato, L. Verardi, Bases of certain finite groups, Ann. Math. Blaise Pascal 1 (1994) 85-93. • R. Scapellato, L. Verardi, Groupes finis qui jouissent d’une propriété analogue au théorème des bases de Burnside., Boll. Unione Mat. Ital. A (7) 5 (1991), 187–194. A. Stocka On sets of generators of finite groups 7 / 28 Matroid groups If a group G has property B, then the cardinality of any g-base of G will be denoted by d(G). A. Stocka On sets of generators of finite groups 8 / 28 Matroid groups If a group G has property B, then the cardinality of any g-base of G will be denoted by d(G). Theorem 6 ( R. Scapellato, L. Verardi, 1991) Let G be a Frattini-free matroid group. If H is a proper subgroup of G, then H has property B, A. Stocka On sets of generators of finite groups 8 / 28 Matroid groups If a group G has property B, then the cardinality of any g-base of G will be denoted by d(G). Theorem 6 ( R. Scapellato, L. Verardi, 1991) Let G be a Frattini-free matroid group. If H is a proper subgroup of G, then H has property B, and d(H) < d(G). A. Stocka On sets of generators of finite groups 8 / 28 Matroid groups Theorem 7 ( R. Scapellato, L. Verardi, 1991) Let G be a Frattini-free group. The group G is a matroid group if and only if one of the following holds: 1. G is an elementary abelian p-group for some prime p, 2. G = P o Q is a scalar extension for primes p 6= q, where q|(p − 1) and Q is cyclic of order q. A. Stocka On sets of generators of finite groups 9 / 28 Scalar extensions Theorem 8 If G is a group such that the quotient group G/Φ(G) is isomorphic to a scalar extension V oϕ H then: 1. G has a unique Sylow p-subgroup P, 2. G is isomorphic to P o Q for any Sylow q-subgroup Q and all Sylow q-subgroups of G are cyclic, m m 3. Φ(G) = Φ(P ) × hxq i where hxq i is the subgroup of index q m in Q = hxi. A. Stocka On sets of generators of finite groups 10 / 28 B-groups Theorem 9 (J. McDougall-Bagnall, M.Quick, 2011) 1. A direct product G1 × G2 of two non-trivial groups G1 and G2 has property B if and only if both G1 and G2 are p-groups for some prime p. A. Stocka On sets of generators of finite groups 11 / 28 B-groups Theorem 9 (J. McDougall-Bagnall, M.Quick, 2011) 1. A direct product G1 × G2 of two non-trivial groups G1 and G2 has property B if and only if both G1 and G2 are p-groups for some prime p. 2. No non-abelian simple groups has property B. A. Stocka On sets of generators of finite groups 11 / 28 B-groups Theorem 9 (J. McDougall-Bagnall, M.Quick, 2011) 1. A direct product G1 × G2 of two non-trivial groups G1 and G2 has property B if and only if both G1 and G2 are p-groups for some prime p. 2. No non-abelian simple groups has property B. Proof. (2) It is well known (via the Classification of Finite Simple Groups) that every non-abelian finite simple group G is 2-generated. On the other hand, if X is the set of involutions in G, then G = hXi and some subset X0 of X is a g-base of G. Necessarily, |X0 | ≥ 3, since a group generated by two involutions is dihedral. Consequently, no non-abelian finite simple group has property B. A. Stocka On sets of generators of finite groups 11 / 28 B-groups Theorem 10 (P. Apisa, B. Klopsch, 2014) Let G be a group with property B. Then: A. Stocka On sets of generators of finite groups 12 / 28 B-groups Theorem 10 (P. Apisa, B. Klopsch, 2014) Let G be a group with property B. Then: 1. every homomorphic image of G has property B; A. Stocka On sets of generators of finite groups 12 / 28 B-groups Theorem 10 (P. Apisa, B. Klopsch, 2014) Let G be a group with property B. Then: 1. every homomorphic image of G has property B; 2. G is soluble; A. Stocka On sets of generators of finite groups 12 / 28 B-groups Theorem 11 (P. Apisa, B. Klopsch, 2014) Let G be a group. Then G is a Frattini-free group with property B if and only if one of the following holds: A. Stocka On sets of generators of finite groups 13 / 28 B-groups Theorem 11 (P. Apisa, B. Klopsch, 2014) Let G be a group. Then G is a Frattini-free group with property B if and only if one of the following holds: 1. G is an elementary abelian p-groups for some prime p; A. Stocka On sets of generators of finite groups 13 / 28 B-groups Theorem 11 (P. Apisa, B. Klopsch, 2014) Let G be a group. Then G is a Frattini-free group with property B if and only if one of the following holds: 1. G is an elementary abelian p-groups for some prime p; 2. G is a scalar extension. A. Stocka On sets of generators of finite groups 13 / 28 B-groups Theorem 12 (P. Apisa, B. Klopsch, 2014) Let G be a group. Then G is a group with property B if and only if one of the following holds: A. Stocka On sets of generators of finite groups 14 / 28 B-groups Theorem 12 (P. Apisa, B. Klopsch, 2014) Let G be a group. Then G is a group with property B if and only if one of the following holds: 1. G is a p-groups for some prime p; A. Stocka On sets of generators of finite groups 14 / 28 B-groups Theorem 12 (P. Apisa, B. Klopsch, 2014) Let G be a group. Then G is a group with property B if and only if one of the following holds: 1. G is a p-groups for some prime p; 2. G = P o Q, where P is a p-group and Q is a cyclic q-group, for distinct primes p 6= q such that CQ (P ) 6= Q and G/Φ(G) is a scalar extension. A. Stocka On sets of generators of finite groups 14 / 28 Groups with the basis property Definition 13 We say that a group G has the basis property if all its subgroups are B-groups. A. Stocka On sets of generators of finite groups 15 / 28 Groups with the basis property Definition 13 We say that a group G has the basis property if all its subgroups are B-groups. • G. Higman, Finite groups in which every element has prime power order, J. London Math. Soc. 32 (1957) 335–342. • P.R. Jones, Basis properties for inverse semigroups, J. Algebra 50 (1978) 135-152. A. Stocka On sets of generators of finite groups 15 / 28 Groups with the basis property Theorem 14 (P.R. Jones, 1978) Let G be a group with the basis property. Then: 1. every element of G is prime power order; A. Stocka On sets of generators of finite groups 16 / 28 Groups with the basis property Theorem 14 (P.R. Jones, 1978) Let G be a group with the basis property. Then: 1. every element of G is prime power order; 2. G is soluble; A. Stocka On sets of generators of finite groups 16 / 28 Groups with the basis property Theorem 14 (P.R. Jones, 1978) Let G be a group with the basis property. Then: 1. every element of G is prime power order; 2. G is soluble; 3. every homomorphic image of G has the basis property; A. Stocka On sets of generators of finite groups 16 / 28 Groups with the basis property Theorem 14 (P.R. Jones, 1978) Let G be a group with the basis property. Then: 1. every element of G is prime power order; 2. G is soluble; 3. every homomorphic image of G has the basis property; 4. if G = G1 × G2 where Gi are nontrivial, then G is a p-group. A. Stocka On sets of generators of finite groups 16 / 28 Groups with the basis property Theorem 14 (P.R. Jones, 1978) Let G be a group with the basis property. Then: 1. every element of G is prime power order; 2. G is soluble; 3. every homomorphic image of G has the basis property; 4. if G = G1 × G2 where Gi are nontrivial, then G is a p-group. A. Stocka On sets of generators of finite groups 16 / 28 Groups with the basis property Theorem 15 (J. McDougall-Bagnall, M.Quick, 2011; J.Krempa, AS, 2014) Let G be a group. Then G has the basis property if and only if the following conditions are satisfied: A. Stocka On sets of generators of finite groups 17 / 28 Groups with the basis property Theorem 15 (J. McDougall-Bagnall, M.Quick, 2011; J.Krempa, AS, 2014) Let G be a group. Then G has the basis property if and only if the following conditions are satisfied: 1. every element in G has prime power order; A. Stocka On sets of generators of finite groups 17 / 28 Groups with the basis property Theorem 15 (J. McDougall-Bagnall, M.Quick, 2011; J.Krempa, AS, 2014) Let G be a group. Then G has the basis property if and only if the following conditions are satisfied: 1. every element in G has prime power order; 2. G is a semidirect product P o Q, where P is a p-group and Q is a cyclic q-group, for some primes p 6= q; A. Stocka On sets of generators of finite groups 17 / 28 Groups with the basis property Theorem 15 (J. McDougall-Bagnall, M.Quick, 2011; J.Krempa, AS, 2014) Let G be a group. Then G has the basis property if and only if the following conditions are satisfied: 1. every element in G has prime power order; 2. G is a semidirect product P o Q, where P is a p-group and Q is a cyclic q-group, for some primes p 6= q; 3. for every subgroup H ≤ G the quotient H/Φ(H) is a scalar extension of (P ∩ H)/Φ(H) by a Sylow q-subgroup of H/Φ(H). A. Stocka On sets of generators of finite groups 17 / 28 Groups with the basis property Theorem 15 (J. McDougall-Bagnall, M.Quick, 2011; J.Krempa, AS, 2014) Let G be a group. Then G has the basis property if and only if the following conditions are satisfied: 1. every element in G has prime power order; 2. G is a semidirect product P o Q, where P is a p-group and Q is a cyclic q-group, for some primes p 6= q; 3. for every subgroup H ≤ G the quotient H/Φ(H) is a scalar extension of (P ∩ H)/Φ(H) by a Sylow q-subgroup of H/Φ(H). A. Stocka On sets of generators of finite groups 17 / 28 Definitions Definition 16 An element g ∈ G will be called a pp-element, if it is of prime power order. A. Stocka On sets of generators of finite groups 18 / 28 Definitions Definition 16 An element g ∈ G will be called a pp-element, if it is of prime power order. Definition 17 A subset X ⊆ G is said to be: • pp-independent if X is a set of pp-elements and is g-independent; A. Stocka On sets of generators of finite groups 18 / 28 Definitions Definition 16 An element g ∈ G will be called a pp-element, if it is of prime power order. Definition 17 A subset X ⊆ G is said to be: • pp-independent if X is a set of pp-elements and is g-independent; • a pp-generating set if X is a set of pp-elements and is a generating set; A. Stocka On sets of generators of finite groups 18 / 28 Definitions Definition 16 An element g ∈ G will be called a pp-element, if it is of prime power order. Definition 17 A subset X ⊆ G is said to be: • pp-independent if X is a set of pp-elements and is g-independent; • a pp-generating set if X is a set of pp-elements and is a generating set; • a pp-base of G, if X is a pp-independent generating set of G. A. Stocka On sets of generators of finite groups 18 / 28 Definitions Definition 16 An element g ∈ G will be called a pp-element, if it is of prime power order. Definition 17 A subset X ⊆ G is said to be: • pp-independent if X is a set of pp-elements and is g-independent; • a pp-generating set if X is a set of pp-elements and is a generating set; • a pp-base of G, if X is a pp-independent generating set of G. A. Stocka On sets of generators of finite groups 18 / 28 Definitions A. Stocka On sets of generators of finite groups 19 / 28 Definitions Definition 18 A group G • has property Bpp (is a Bpp -group) if all pp-bases of G have the same size ; A. Stocka On sets of generators of finite groups 20 / 28 Definitions Definition 18 A group G • has property Bpp (is a Bpp -group) if all pp-bases of G have the same size ; • has the pp-embedding property if every pp-independent subset of G is contained in a pp-base of G; A. Stocka On sets of generators of finite groups 20 / 28 Definitions Definition 18 A group G • has property Bpp (is a Bpp -group) if all pp-bases of G have the same size ; • has the pp-embedding property if every pp-independent subset of G is contained in a pp-base of G; • is pp-matroid if G has property Bpp and the pp-embedding property. A. Stocka On sets of generators of finite groups 20 / 28 Groups with pp-basis property Lemma 19 Let G be a group. Then: A. Stocka On sets of generators of finite groups 21 / 28 Groups with pp-basis property Lemma 19 Let G be a group. Then: 1. Every pp-independent set of G/Φ(G) can be lifted to a pp-independent set of G. A. Stocka On sets of generators of finite groups 21 / 28 Groups with pp-basis property Lemma 19 Let G be a group. Then: 1. Every pp-independent set of G/Φ(G) can be lifted to a pp-independent set of G. 2. G is a Bpp -group if and only if G/Φ(G) is a Bpp -group. Definition 20 We say that a group G has the pp-basis property if all its subgroups are Bpp -groups. A. Stocka On sets of generators of finite groups 21 / 28 References • J. Krempa, A. Stocka, On some sets of generators of finite groups, J. Algebra 405 (2014), 122-134. • J. Krempa, A. Stocka, Corrigendum to “On some sets of generators of finite groups”, J. Algebra 408 (2014), 61-62. • J. Krempa, A. Stocka, On sets of pp-generators of finite groups, Bull. Aust. Math. Soc. 91 (2015), 241–249. A. Stocka On sets of generators of finite groups 22 / 28 Groups with the pp-basis property Theorem 21 (J. Krempa, AS, 2014) Let G1 and G2 be groups with coprime orders. 1. G1 and G2 are B pp -groups if and only if G1 × G2 is a B pp -group. 2. G1 and G2 have the pp-basis property if and only if G1 × G2 has the pp-basis property. A. Stocka On sets of generators of finite groups 23 / 28 Groups with the pp-basis property Theorem 21 (J. Krempa, AS, 2014) Let G1 and G2 be groups with coprime orders. 1. G1 and G2 are B pp -groups if and only if G1 × G2 is a B pp -group. 2. G1 and G2 have the pp-basis property if and only if G1 × G2 has the pp-basis property. Example 22 Let p 6= q be primes and q|(p − 1), for suitable 1 < r < p G1 = hx, y | xp = y q = 1, xy = xr i and G2 = hz | z q = 1i. Then the sets {x, y, z} and {x, yz} are pp-bases of G1 × G2 . It follows that G1 × G2 has not property B pp and so has not the pp-basis property. However G1 and G2 are groups with pp-basis property. A. Stocka On sets of generators of finite groups 23 / 28 Groups with the pp-basis property Theorem 23 (J. Krempa, AS, 2014) Let G be a group and H ≤ G be a normal subgroup. 1. If G is a Bpp -group, then G/H is also a Bpp -group. 2. If G has the pp-basis property, then G is soluble and G/H has the pp-basis property. A. Stocka On sets of generators of finite groups 24 / 28 ’ A. Stocka On sets of generators of finite groups 25 / 28 Groups with the pp-basis property Definition 24 A group G will be called (coprimely) indecomposable if it is not a direct product of nontrivial groups with coprime orders. Theorem 25 (J. Krempa, AS, 2015) Let G be an indecomposable {p, q}-group with the pp-basis property. Then G = P o Q, where P is a p-group and Q is cyclic q-group for primes p 6= q and for every H ≤ G either H/Φ(H) is a scalar extension or H = PH × QH , where PH = P ∩ H and QH is a Sylow q-subgroup of H. A. Stocka On sets of generators of finite groups 25 / 28 Groups with the pp-basis property Theorem 26 (J. Krempa, AS, 2015) Let G be a group. Then G has the pp-basis property if and only if it is one of the following groups: 1. a p-group; A. Stocka On sets of generators of finite groups 26 / 28 Groups with the pp-basis property Theorem 26 (J. Krempa, AS, 2015) Let G be a group. Then G has the pp-basis property if and only if it is one of the following groups: 1. a p-group; 2. an indecomposable {p, q}-group with the pp-basis property; A. Stocka On sets of generators of finite groups 26 / 28 Groups with the pp-basis property Theorem 26 (J. Krempa, AS, 2015) Let G be a group. Then G has the pp-basis property if and only if it is one of the following groups: 1. a p-group; 2. an indecomposable {p, q}-group with the pp-basis property; 3. a direct product of groups given in (1) and (2) with pairwise coprime orders. A. Stocka On sets of generators of finite groups 26 / 28 pp-Maroid groups Proposition 27 (J. Krempa, AS, 2015) Let G be a Frattini-free pp-matroid group. If H is a proper subgroup of G, then H is a Bpp -group and dpp (H) < dpp (G). A. Stocka On sets of generators of finite groups 27 / 28 pp-Maroid groups Proposition 27 (J. Krempa, AS, 2015) Let G be a Frattini-free pp-matroid group. If H is a proper subgroup of G, then H is a Bpp -group and dpp (H) < dpp (G). Theorem 28 (J. Krempa, AS, 2015) Let G be a Frattini-free group. The group G is a pp-matroid group if and only if one of the following holds: 1. G is a p-group for some prime p, 2. H = P o Q is a scalar extension for primes p 6= q, where q|(p − 1) and Q is cyclic of order q, 3. G is a direct product of groups given in (1) and (2) with coprime orders. A. Stocka On sets of generators of finite groups 27 / 28 THANK YOU FOR YOUR ATTENTION A. Stocka On sets of generators of finite groups 28 / 28