On some numerical invariants of finite groups Jan Krempa Institute - PDF document

On some numerical invariants of finite groups Jan Krempa Institute of Mathematics, University of Warsaw ul. Banacha 2, 02-097 Warszawa, Poland jkrempa@mimuw.edu.pl ( with Agnieszka Stocka)

References [1] P.R. Jones, Basis properties for inverse semigroups, J. Algebra 50(1978) 135-152. [2] J. Krempa, A. Stocka, On some invariants of finite groups, Int. J. Group Theory 2(1)(2013) 109-115. [3] J. McDougall-Bagnall, M. Quick, Groups with the basis property, J. Algebra 346(2011), 332-339. [4] P. Apisa, B. Klopsch, Groups with a base property analogous to that of vector spaces, arXiv 1211:6137v1 [math.GR] 26 Nov 2012. [5] D. Levy, (Personal communica- tion). 0

1 Preliminaries All groups, usually G, are finite. A numerical invariant of G is a non- negative 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

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 nu- merical invariant is also the covering number. We did not consider its mo- notonicity. 2

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

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 1 1 · . . . · p k 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

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

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 prop- erty. 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 nontriv- ial elements in G are p -groups, ( G is a CP-group) . 6

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

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 con- tains 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

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 con- sider an action φ i : Q − → Aut K V via ‘scalar’ multiplication: → vρ j x j φ i : v − i , (7) and the semidirect product G i = V ⋊ φ i Q with the above mentioned action. Then G i has prop- erty B . 9

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. How- ever the F p [ Q ]-module structures on V induced by φ i and φ j are nonisomorphic if V � = 0 and i � = j. 10

Example 2.4 (Diagonal extension) . Under the notation from Lemma 2.2 and previous exam- ple 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: → ( v 1 ρ j x j ϕ : ( v 1 + . . . + v s ) − 1 + . . . + v s ρ 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

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 sub- group 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

3 Some results Proposition 3.1. Let G = P ⋊ Q be a semidi- rect 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 fixed- point-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

Corollary 3.2. Let G = V ⋊ ϕ Q be a diago- nal 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 exten- sion 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

Under the above terminology, a cor- rected 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 fol- lowing 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

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 x j = a 2 j b x j = b 2 j and for 1 ≤ j ≤ 3 . Thus, c x j = c 2 2 j = c 4 j . 16

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

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 P 1 ≃ ( F 49 ) + m = 1 , K = F 7 , and the action of Q is not linear over F 49 . 18

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 prop- erty. In this group, sg ( G ) = ig ( G ) = 2 , but sg ( P ) = ig ( P ) = 3 . Hence, neither ig nor sg is a monotone invari- ant. 19