Central Units in Metacyclic Integral Group Rings

Ferraz, Raul Antonio; Simón-Pınero, Juan Jacobo

doi:10.1080/00927870802158028

Cited by 16 publications

(18 citation statements)

References 5 publications

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“…Here b (n) is defined recursively as follows. We denote by Z i the i-th centre of G. Recently, Ferraz and Simón [2] gave a construction of central units in case G is meta-(cyclic of prime order) that generate a subgroup of finite index. They actually provided an independent set of generators.…”

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confidence: 99%

Construction of central units in integral group rings of finite groups

Jespers

Parmenter

2011

Proc. Amer. Math. Soc.

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Abstract. In this paper we give new constructions of central units that generate a subgroup of finite index in the central units of the integral group ring ZG of a finite group. This is done for a very large class of finite groups G, including the abelian-by-supersolvable groups.Let G be a finite group. When G is abelian it is well known (and due to Bass and Milnor; see [1] and [9, Theorem 12.7 and Theorem 13.1]) that the Bass cyclic units of the integral group ring ZG generate a subgroup of finite index in Z(U(ZG)), the group of (central) units of ZG. In [4], it was shown that if G is a nilpotent finite group of nilpotency class n, then the group b (n) is of finite index in Z(U(ZG)).Here b (n) is defined recursively as follows. We denote by Z i the i-th centre of G. For any x ∈ G and Bass cyclic unit b ∈ Z x , put b (1) = b, and, for 2 ≤ i ≤ n, putRecently, Ferraz and Simón [2] gave a construction of central units in case G is meta-(cyclic of prime order) that generate a subgroup of finite index. They actually provided an independent set of generators.In this note we continue these investigations and provide new constructions of central units that generate a subgroup of finite index in the central units of ZG and this for a very large class of finite groups G, including the abelian-by-supersolvable groups. The proof and construction relies on a beautiful paper of Olivieri, del Río and Simón [6] in which they give an explicit construction of the primitive central idempotents of rational group algebras QG of the so-called strongly monomial groups (for example, abelian-by-supersolvable groups). Our proof thus uses a completely different method from the one used in [4] for finite nilpotent groups. In Section 1 we will recall the necessary background on this. In Section 2 we prove the main results for strongly monomial groups. In Section 3 we prove a general result that yields generators for central units from subgroups and quotient groups. An application is given for solvable Frobenius groups.

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confidence: 99%

Construction of central units in integral group rings of finite groups

Jespers

Parmenter

2011

Proc. Amer. Math. Soc.

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“…However, we obtained an explicit description of a virtual basis of Z(U(ZG)) when G is a finite abelian-by-supersolvable group (and thus a strongly monomial group) such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in G. Note that the latter does not apply to all finite split metacyclic groups C m ⋊ C n , for example if n is a prime number and C m ⋊ C n is not abelian then C n is not subnormal in C m ⋊ C n . On the other hand, Ferraz and Simón did construct in [9] a virtual basis of Z(U(Z(C q ⋊ C p ))) for p and q odd and different primes. In our second main result (Theorem 3.5) we extend these results on the construction of a virtual basis of Z(U(ZG)) to a class of finite strongly monomial groups containing the metacyclic groups G = C q m ⋊ C p n with p and q different primes and C p n acting faithfully on C q m .…”

Section: Introductionmentioning

confidence: 99%

“…Our proof makes again use of strong Shoda pairs and the description of the Wedderburn decomposition of QG obtained by Olivieri, del Río and Simón in [3]. Our approach is thus different from the one used in [9].…”

Section: Introductionmentioning

confidence: 99%

Group rings of finite strongly monomial groups: Central units and primitive idempotents

et al. 2013

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We compute the rank of the group of central units in the integral group ring ZG of a finite strongly monomial group G. The formula obtained is in terms of the strong Shoda pairs of G. Next we construct a virtual basis of the group of central units of ZG for a class of groups G properly contained in the finite strongly monomial groups. Furthermore, for another class of groups G inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of QG.Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of ZG, this for metacyclic groups G of the form G = C q m ⋊ C p n with p and q different primes and the cyclic group C p n of order p n acting faithfully on the cyclic group C q m of order q m .

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“…The structure of Z(U(Z[G])) has been a * Research supported by IISER, Mohali, India, is gratefully acknowledged † Corresponding author subject of intensive research (see e.g. [4,10,11,14,15,16,17,18,19,25]). Clearly Z(U(Z[G])) contains the so-called trivial central units ±g, g ∈ Z(G), the centre of G. Thus there arises the problem of characterizing the groups G having the property that all central units of the integral group ring Z[G] are trivial.…”

Section: Introductionmentioning

confidence: 99%