The Multiplicative Jordan Decomposition in Group Rings

Arora, Satya R.; Hales, Alfred W.; Passi, Inder Bir S.

doi:10.1006/jabr.1998.7557

Cited by 16 publications

(5 citation statements)

References 6 publications

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“…Usually, α s and α u do not always lie in Z [G]. Following [AHP98] and [HPW07], we say that Z[G] has the multiplicative Jordan decomposition property (MJD) if for every unit α of Z[G], those α s and α u are contained in Z [G].…”

Section: Introductionmentioning

confidence: 99%

The Multiplicative Jordan Decomposition in the Integral Group Ring $\mathbb{Z}[Q_8 \times C_p]$

Kuo,

Sun

2020

Preprint

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Let p be a prime such that the multiplicative order m of 2 modulo p is even. We prove that the integral group ring Z[Q 8 × C p ] has the multiplicative Jordan decomposition property when m is congruent to 2 modulo 4. There are infinitely many such primes and these primes include the case p ≡ 3 (mod 4). We also prove that Z[Q 8 × C 5 ] has the multiplicative Jordan decomposition property in a new way.

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Section: Introductionmentioning

confidence: 99%

The Multiplicative Jordan Decomposition in the Integral Group Ring $\mathbb{Z}[Q_8 \times C_p]$

Kuo,

Sun

2020

Preprint

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“…This is the unique multiplicative Jordan decomposition of a. Following [2] and [5], we say that Z[G] has the multiplicative Jordan decomposition property (MJD) if for every unit a of Z[G], its semisimple and unipotent parts are both contained in Z [G]. For simplicity, we say that G satisfies MJD if its integral group ring Z[G] has this property.…”

Section: Introductionmentioning

confidence: 99%

“…In the non-Dedekind case, it appears that the MJD property is relatively rare. Indeed, the papers [2] and [5] have shown that Z[G] and Q[G] must be quite restrictive. For example, we have the following, with part (i) from [2,Theorem 4.1] and part (ii) from [5,Corollary 9].…”

Section: Introductionmentioning

confidence: 99%

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Multiplicative Jordan Decomposition in Group Rings of 2, 3-Groups

Liu

Passman

2010

J. Algebra Appl.

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In this paper, we essentially finish the classification of those finite 2, 3-groups G having integral group rings with the multiplicative Jordan decomposition (MJD) property. If G is abelian or a Hamiltonian 2-group, then it is clear that ℤ[G] satisfies MJD. Thus, we need only consider the nonabelian case. Recall that the 2-groups with MJD were completely determined by Hales, Passi and Wilson, while the corresponding 3-groups were almost completely determined by the present authors. Thus, we are concerned here, for the most part, with groups whose order is divisible by 6. As it turns out, there are precisely three nonabelian 2, 3-groups, of order divisible by 6, with ℤ[G] satisfying MJD. These have orders 6, 12, and 24. In view of another result of Hales, Passi and Wilson, this completes a significant portion of the classification of all finite groups with MJD.

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“…Indeed, the papers [AHP] and [HPW] have shown that Z[G] and Q[G] must be quite restrictive. For example, we have the following, with part (i) from [AHP,Theorem 4.1] and part (ii) from [HPW,Corollary 9]. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Multiplicative Jordan Decomposition in Group Rings of 3-Groups

LIU

Passman

2009

J. Algebra Appl.

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In this paper, we essentially classify those finite 3-groups G having integral group rings with the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ℤ[G] satisfies MJD. Thus, we are only concerned with the nonabelian case. Here we show that ℤ[G] has the MJD property for the two nonabelian groups of order 33. Furthermore, we show that there are at most three other specific nonabelian groups, all of order 34, with ℤ[G] having the MJD property. Unfortunately, we are unable to decide which, if any, of these three satisfies the appropriate condition.

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The Multiplicative Jordan Decomposition in Group Rings

Cited by 16 publications

References 6 publications

The Multiplicative Jordan Decomposition in the Integral Group Ring $\mathbb{Z}[Q_8 \times C_p]$

The Multiplicative Jordan Decomposition in the Integral Group Ring $\mathbb{Z}[Q_8 \times C_p]$

Multiplicative Jordan Decomposition in Group Rings of 2, 3-Groups

Multiplicative Jordan Decomposition in Group Rings of 3-Groups

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