Non-abelian tensor and exterior products modulo q and universal q-central relative extension

Conduché, Daniel; Rodríguez-Fernández, C.

doi:10.1016/0022-4049(92)90092-t

Cited by 19 publications

(11 citation statements)

References 12 publications

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“…Theorem 6 and Proposition 7 in [4] (the first of these results being due to [3]) yield the following diagram of solid homomorphisms:…”

Section: Moreover There Is a Groupmentioning

confidence: 96%

A bound for the derived and Frattini subgroups of a prime-power group

Ellis

1998

Proc. Amer. Math. Soc.

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Abstract. This paper is based on the seemingly new observation that the Schur multiplier M (G) of a d-generator group of prime-power order p n has order |M (G)| ≤ p d(2n−d−1)/2 . We prove several related results, including sufficient conditions for a sharper bound on |M (G)| to be an equality.Let G be a group with centre Z. Schur [13] observed that the commutator subgroup [G, G] is finite whenever the quotient G/Z is finite. Wiegold [17] obtained an estimate for the order of [G, G] in terms of the order of G/Z; in particular he showed that if G/Z has prime-power order p n , then the order of [G, G] is at most p n(n−1)/2 . This bound can be attained for all n ≥ 1, but only if G/Z is elementary abelian. As a corollary, Wiegold re-derived Green's bound [7] on the order of the Schur multiplier of a prime-power group. Gaschütz, Neubüser and Yen In this article we reduce the above bounds on [G, G] by incorporating, in turn: (1) the number of generators of G/Z; (2) the abelian group structure of the quotients of the lower central series of G/Z; (3) the restricted Lie algebra structure of the quotients of the lower p-central series of G/Z; (4) the "breadth in G" of the preimages of the generators of G/Z. The fourth approach (which is the only one to involve more than the structure of G/Z) complements a result of Vaughan-Lee [16]. We obtain corresponding reductions in the above-mentioned bounds on the Schur multiplier. As an application we obtain a rough upper bound on the number of d-generator groups of order p n . Furthermore, our results are presented in such a way as to yield bounds on the Frattini subgroup of a prime-power group.Let p be any prime, and let q ≥ 0 be any nonnegative integer multiple of p. (We are interested mainly in the cases q = 0 and q = p.) Given a group G we let Z q (G) denote the subgroup of the centre of G consisting of those elements with order dividing q. Given a normal subgroup N in G, we let N# q G denote the subgroup of G generated by the commutators ngn −1 g −1 and powers n q for n ∈ N , g ∈ G.

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“…Theorem 6 and Proposition 7 in [4] (the first of these results being due to [3]) yield the following diagram of solid homomorphisms:…”

Section: Moreover There Is a Groupmentioning

confidence: 96%

A bound for the derived and Frattini subgroups of a prime-power group

Ellis

1998

Proc. Amer. Math. Soc.

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“…Let G and H be normal subgroups of some group L and q a non-negative integer. The definition of the q-tensor product, G⊗ q H, of G and H has evolved in papers [15,2,7]. Let K = { k} be a set of symbols, one for each k ∈ G ∩ H (If q = 0 then K is taken to be the empty set).…”

Section: Q-tensor Productmentioning

confidence: 99%

“…The existence and the structure of universal q-central extensions were studied by Brown in [2] (see also Conduché and Rodríguez-Fernández [7]). By using Theorem 5.1, Brown proved that if G is a q-perfect group, that is, G = G ′ G q , where G q is the subgroup of G generated by the set {g q |g ∈ G}, then universal q-central extensions of G are isomorphic to the sequence…”

Section: Q-tensor Productmentioning

confidence: 99%

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A survey of non-abelian tensor products of groups and related constructions

Nakaoka

Rocco²

2012

B Soc Paran Mat

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“…It is a well known fact (see [23]) that the subgroup Υ(G) = [G, G ϕ ] of ν(G) is isomorphic to the non-abelian tensor square G ⊗ G, as defined by Brown and Loday in their seminal paper [8]. A modular version of the operator ν was considered in [10], where for any non-negative integer q the authors introduced and studied a group ν q (G), which in turn is an extension of the so called q-tensor square of G, G⊗ q G, first defined by Conduché and Rodriguez-Fernandez in [11] (see also [14], [7]). In order to describe the group ν q (G), if q ≥ 1 then let G = { k | k ∈ G} be a set of symbols, one for each element of G (for q = 0 we set G = ∅, the empty set).…”

Section: Introductionmentioning

confidence: 99%

The q-tensor square of finitely generated nilpotent groups, q odd

Rocco

Rodrigues

2017

J. Algebra Appl.

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In the present paper the authors extend to the q−tensor square G ⊗ q G of a group G, q a non-negative integer, some structural results due to R. D. Blyth, F. Fumagalli and M. Morigi concerning the non-abelian tensor square G⊗G (q = 0). The results are applied to the computation of G⊗ q G for finitely generated nilpotent groups G, specially for free nilpotent groups of finite rank. We also generalize to all q ≥ 0 results of M. Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square G ⊗ G when G is a n−generator nilpotent group of class 2. We end by computing the q−tensor squares of the free n−generator nilpotent group of class 2, n ≥ 2, for all q ≥ 0. This shows that the above mentioned upper bound is also achieved for these groups when q > 1.

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Non-abelian tensor and exterior products modulo q and universal q-central relative extension

Cited by 19 publications

References 12 publications

A bound for the derived and Frattini subgroups of a prime-power group

A bound for the derived and Frattini subgroups of a prime-power group

A survey of non-abelian tensor products of groups and related constructions

The q-tensor square of finitely generated nilpotent groups, q odd

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