Higman’s Central Unit Theory, Units of Integral Group Rings of Finite Cyclic Groups and Fibonacci Numbers

Aleev, R. Zh.

doi:10.1142/s0218196794000038

Cited by 18 publications

(29 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…If n ≡ 0 (mod 4), a similar argument shows that the cases n = 4, 8, 12 cannot occur since µ = [7,5,3,1] is the least partition that can be written as a sum of at least four odd pairwise different numbers. If n = 9, then there exist only two partitions which satisfy the first two conditions of Theorem 4.5, namely [5,3,1] and [9]. The partition [5,3,1] does not satisfy the third condition, and [9] does not satisfy the fourth one.…”

Section: Proof Letmentioning

confidence: 92%

“…Let τ ∈ A n be a cycle of odd length m. Write m = p r 1 1 · · · p r a a , with p i rational primes, 1 i a. Then Z m ∼ = τ = (a 1 , a 2 , .…”

Section: The Rank Of Z(u (Z Z Za N ))mentioning

confidence: 99%

“…Let g = g 1 · · · g k be the decomposition of g as a product of disjoint cycles, with g j the cycle of order µ j . Let m = p r 1 1 · · · p r a a be the decomposition of m as a product of distinct primes and let r i,j be the exponent of p i in the decomposition of µ j , 1 j k. Also let t i , 1 i a, be elements as defined right after the proof of Proposition 4.2.…”

Section: Proof Letmentioning

confidence: 99%

See 2 more Smart Citations

Simple components and central units in group algebras

Ferraz

2004

Journal of Algebra

View full text Add to dashboard Cite

Let G be a finite group. Berman [Dokl. Akad. Nauk 106 (1956) 767] and Witt [J. Reine Angew. Math. 190 (1952) 231] evaluate, independently, the number of simple components of the group algebra F G when F is a field of characteristic 0. In this paper we extend this result to fields of arbitrary characteristic which does not divide the order of G. We also compute the rank of the group of the central units of ZG and obtain an alternative proof of a well-known result of Ritter and Sehgal.

show abstract

Section: Proof Letmentioning

confidence: 92%

“…Let τ ∈ A n be a cycle of odd length m. Write m = p r 1 1 · · · p r a a , with p i rational primes, 1 i a. Then Z m ∼ = τ = (a 1 , a 2 , .…”

Section: The Rank Of Z(u (Z Z Za N ))mentioning

confidence: 99%

Section: Proof Letmentioning

confidence: 99%

See 1 more Smart Citation

Simple components and central units in group algebras

Ferraz

2004

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…In order to study Z(U(Z[G])), a multiplicatively independent subset of such a subgroup A, i.e., a Z-basis for such a free Z-module A, is of importance, and is known only for a few groups ( [1,2,7,18], see also [20], Examples 8.3.11 and 8.3.12). However, other papers deal with determining a virtual basis of Z(U(Z[G])),…”

Section: Let U(z[g]) Denote the Unit Group Of The Integral Group Ringmentioning

confidence: 99%

On the index of a free abelian subgroup in the group of central units of an integral group ring

Bakshi

Maheshwary

2015

Journal of Algebra

View full text Add to dashboard Cite

Let Z(U (Z[G])) denote the group of central units in the integral group ring Z[G] of a finite group G. A bound on the index of the subgroup generated by a virtual basis in Z(U (Z[G])) is computed for a class of strongly monomial groups. The result is illustrated with application to the groups of order p n , p prime, n ≤ 4. The rank of Z(U (Z[G])) and the Wedderburn decomposition of the rational group algebra of these p-groups have also been obtained.

show abstract

“…The construction depended on the existence of a finite normal series in G. If we choose A n (n > 4) as a finite group, it is impossible to construct generators for U(Z(ZG)) from Bass cyclic units since A n is a simple group. However Aleev [1] constructed all central units of ZA 5 and ZA 6 . Later, Li and Parmenter [7] independently constructed all central units of ZA 5 , too.…”

Section: Introductionmentioning

confidence: 99%