On some numerical invariants of finite groups. Jan Krempa Institute of Mathematics, University of Warsaw

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1 On some numerical invariants of finite groups Jan Krempa Institute of Mathematics, University of Warsaw ul. Banacha 2, Warszawa, Poland (with Agnieszka Stocka)

2 References [1] P.R. Jones, Basis properties for inverse semigroups, J. Algebra 50(1978) [2] J. Krempa, A. Stocka, On some invariants of finite groups, Int. J. Group Theory 2(1)(2013) [3] J. McDougall-Bagnall, M. Quick, Groups with the basis property, J. Algebra 346(2011), [4] P. Apisa, B. Klopsch, Groups with a base property analogous to that of vector spaces, arxiv 1211:6137v1 [math.gr] 26 Nov [5] D. Levy, (Personal communication). 0

3 1 Preliminaries All groups, usually G, are finite. A numerical invariant of G is a nonnegative integer α(g), such that G H α(g) = α(h). (1) As in [2] an invariant α is monotone (on G) if α is defined for all subgroups of G and α(h) α(k), whenever H K G are subgroups. An obvious monotone invariant is G, the order of G. G = H G : H. (2) 1

4 Extending ideas from [5], for every nontrivial word w = w(x 1,..., x n ) in a free group, let us consider an invariant s w (G) = (g 1,..., g n ) G n : w(g 1,... g n ) = 1. This is a monotone invariant. An interesting and often studied numerical invariant is also the covering number. We did not consider its monotonicity. 2

5 For every group G let Φ(G) denotes the Frattini subgroup of G. A subset X of G is said here: g-independent if Y, Φ(G) X, Φ(G) for all Y X; a generating set of G if X = G; a g-base of G, if X is a g-independent generating set of G. Every group has a g-base. 3

6 For any group G let ig(g) = inf X X, sg(g) = sup X, (3) X where X runs over all g-bases of G. Theorem 1.1 (Burnside). If G is a p-group with G/Φ(G) = p r, then ig(g) = sg(g) = r. Example 1.2. Let G be cyclic of order n = p k pk r r where p i are distinct primes and k i > 0 for i = 1,..., r. Then sg(g) = r, while ig(g) = 1. 4

7 Proposition 1.3 ([2]). Let G and H be groups with coprime orders. Then: ig(g H) = max(ig(g), ig(h)), (4) while sg(g H) = sg(g) + sg(h). (5) Corollary 1.4. Let 1 m n <. Then there exists a group G such that ig(g) = m and sg(g) = n. 5

8 2 The basis property Groups G with ig(g) = sg(g) are known as groups with property B. Groups with all subgroups having property B are known as groups with the basis property. By Theorem 1.1 p-groups have the basis property. It is well known that in any group G, every element has prime power order if and only if centralizers of nontrivial elements in G are p-groups, (G is a CP-group). 6

9 Theorem 2.1 ([1]). Let G be a group with the basis property. Then: 1. G is a CP-group; 2. G is soluble; 3. Every homomorphic image of G has the basis property; 4. If G = G 1 G 2 where G i are nontrivial, then G is a p-group. Results analogous to claims 2 and 3 above, but for groups with property B can be found in [4]. 7

10 Lemma 2.2. Let p q be primes and m 0. Then there exists the smallest field K = K(p, q, m) of characteristic p such that K contains all q m -th roots of 1 K. If ρ 1,..., ρ s are all primitive q m -th roots of 1 in K, then K = F p [ρ 1 ] =... = F p [ρ s ], (6) where F p is the prime field with p elements. Also, s = (q 1)q m 1 for m 1 and s = 1 for m = 0. 8

11 Example 2.3 (Scalar extension, see [3]). For p, q, m, s and K as above, let Q = x be a cyclic group of order q m. If V is a vector space over K then, for every 1 i s we can consider an action φ i : Q Aut K V via scalar multiplication: x j φ i : v vρ j i, (7) and the semidirect product G i = V φi Q with the above mentioned action. Then G i has property B. 9

12 Subgroups of G i are also scalar extensions, possibly with use of smaller fields. Hence, G i has the basis property. The groups G i are pairwise isomorphic. However the F p [Q]-module structures on V induced by φ i and φ j are nonisomorphic if V 0 and i j. 10

13 Example 2.4 (Diagonal extension). Under the notation from Lemma 2.2 and previous example let our K-vector space V be a direct sum V = V 1... V s where V i are K-subspaces. Consider the action ϕ : Q End K (V ) given by a diagonal multiplication: x j ϕ : (v v s ) (v 1 ρ j v sρ j s), (8) where v i V i for i = 1,... s. Let G ϕ = V ϕ Q be the semidirect product with the above mentioned action. 11

14 The subspaces V i will be named Q-components of V. The group G ϕ is a CP-group and Q acts fixed point freely on V. Every Q-invariant subgroup of V is a K-subspace, by Formula (6). If G ϕ is not a scalar extension, then not every K-subspace of V is Q-invariant. Moreover, the group G ϕ has no basis property, and even it has not property B. 12

15 3 Some results Proposition 3.1. Let G = P Q be a semidirect product of an elementary abelian p-group P and a cyclic q-group Q of order q m, where p q are primes, and let K = K(p, q, m) be the field constructed in Lemma 2.2. Then the following conditions are equivalent: (i) Every non-identity element of Q acts fixedpoint-freely on P ; (ii) G is a CP-group; (iii) P is a vector space over K and G is a diagonal extension of P by Q. 13

16 Corollary 3.2. Let G = V ϕ Q be a diagonal extension, for the suitable field K. Then: 1. G is a B-group if and only if G is a scalar extension; 2. Any subgroup H G is a diagonal extension of V H = V H by a Sylow q-subgrop Q H of H, with not more Q H -components of V H than Q-components of V ; 3. G satisfies the basis property if and only if it is a scalar extension. 14

17 Under the above terminology, a corrected version of Theorem 1.1 from [3] reads: Theorem 3.3. Let G be a group. Then G has the basis property if and only if the following conditions are satisfied: 1. G is a CP-group, 2. G P Q, where P is a p-group, Q is a cyclic q-group, for primes q p, 3. For every subgroup H G, H/Φ(H) is a scalar extension of (P H)/φ(H) by a Sylow q-subgroup of H/Φ(H). 15

18 Example 3.4. Consider the group P = a, b a 7 = b 7 = c 7 = 1 = [a, c] = [b, c], where c = [a, b]. Then P = 7 3 and Exp(P ) = 7. Let Q = x be the cyclic group of order 3. Then Q can act on P in the following way: a xj = a 2j and b xj = b 2j for 1 j 3. Thus, c xj = c 22j = c 4j. 16

19 Let G = P Q under the above action. G is a 3-generated CP-group and we have Φ(G) = Φ(P ) = c. Now, G/Φ(G) is a scalar extension of P/Φ(G) hence, G/Φ(G) is a B-group and even has the basis property. Thus also G is a B-group. 17

20 If H = a, c, x then P 1 = a, c = P H is an elementary abelian 7-group, Φ(H) = Φ(P 1 ) = 1 and H is a diagonal, but not a scalar extension of P 1, because 2 4 in F 7. Hence G does not satisfy the basis property. In the above example m = 1, K = F 7, P 1 (F 49 ) + and the action of Q is not linear over F

21 Example 3.5. Under standard notation of this notes, assume that P = 2 3 and Q = 7. Then, any nonabelian P Q = G has the basis property. In this group, sg(g) = ig(g) = 2, but sg(p ) = ig(p ) = 3. Hence, neither ig nor sg is a monotone invariant. 19

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