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

On some numerical invariants

• f 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

• f vector spaces, arXiv 1211:6137v1

[math.GR] 26 Nov 2012. [5] D. Levy, (Personal communica- tion).

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

• f 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)

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Extending ideas from [5], for every nontrivial word w = w(x1, . . . , xn) in a free group, let us consider an invariant sw(G) = |(g1, . . . , gn) ∈ Gn : w(g1, . . . gn) = 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.

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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.

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For any group G let ig(G) = inf

X |X|,

sg(G) = sup

X

|X|, (3) where X runs over all g-bases of G. Theorem 1.1 (Burnside). If G is a p-group with |G/Φ(G)| = pr, then ig(G) = sg(G) = r. Example 1.2. Let G be cyclic of order n = pk1

1 · . . . · pkr r where pi are distinct primes and

ki > 0 for i = 1, . . . , r. Then sg(G) = r, while ig(G) = 1.

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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.

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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).

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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 = G1 × G2 where Gi 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].

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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 qm-th roots of 1 ∈ K. If ρ1, . . . , ρs are all primitive qm-th roots of 1 in K, then K = Fp[ρ1] = . . . = Fp[ρs], (6) where Fp is the prime field with p elements. Also, s = (q − 1)qm−1 for m ≥ 1 and s = 1 for m = 0.

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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 qm. If V is a vector space

• ver K then, for every 1 ≤ i ≤ s we can con-

sider an action φi : Q − → AutKV via ‘scalar’ multiplication: xjφi : v − → vρj

i,

(7) and the semidirect product Gi = V ⋊φi Q with the above mentioned action. Then Gi has prop- erty B.

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Subgroups of Gi are also scalar extensions, possibly with use of smaller fields. Hence, Gi has the basis property. The groups Gi are pairwise isomorphic. How- ever the Fp[Q]-module structures on V induced by φi and φj are nonisomorphic if V = 0 and i = j.

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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 = V1 ⊕ . . . ⊕ Vs where Vi are K-subspaces. Consider the action ϕ : Q − → EndK(V ) given by a ‘diagonal’ multiplication: xjϕ : (v1+. . .+vs) − → (v1ρj

1+. . .+vsρj s), (8)

where vi ∈ Vi for i = 1, . . . s. Let Gϕ = V ⋊ϕ Q be the semidirect product with the above mentioned action.

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The subspaces Vi will be named Q-components

• f V. The group Gϕ 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 Gϕ is not a scalar extension, then not every K-subspace of V is Q-invariant. Moreover, the group Gϕ has no basis property, and even it has not property B.

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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 qm, 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.

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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 VH = V ∩H by a Sylow q-subgrop QH of H, with not more QH-components

• f VH than Q-components of V ;
• 3. G satisfies the basis property if and only

if it is a scalar extension.

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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).

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Example 3.4. Consider the group P = a, b | a7 = b7 = c7 = 1 = [a, c] = [b, c], where c = [a, b]. Then |P| = 73 and Exp(P) =

• 7. Let Q = x be the cyclic group of order 3.

Then Q can act on P in the following way: axj = a2j and bxj = b2j for 1 ≤ j ≤ 3. Thus, cxj = c22j = c4j.

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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.

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If H = a, c, x then P1 = a, c = P ∩ H is an elementary abelian 7-group, Φ(H) = Φ(P1) = 1 and H is a diagonal, but not a scalar extension of P1, because 2 = 4 in F7. Hence G does not satisfy the basis property. In the above example m = 1, K = F7, P1 ≃ (F49)+ and the action of Q is not linear over F49.

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Example 3.5. Under standard notation of this notes, assume that |P| = 23 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.

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