Partial augmentations and jordan decomposition in group ring

Hales, Alfred W.; Luthar, Indar S.

doi:10.1080/00927879008824023

Cited by 18 publications

(9 citation statements)

References 4 publications

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“…To start with, [3,Example 3.3] shows that Z[Sym 3 ] has the additive and hence multiplicative Jordan decomposition property. In fact, by [3,Theorem 3.4], this holds for all dihedral groups of order 2p with p an odd prime. Next, [1, Proposition 3.1] handles the group (ii) of order 12.…”

Section: The Main Resultsmentioning

confidence: 99%

“…Obviously, N is multiplicative and the ordering of the factors in N does not matter. In particular, in Z[G] we have (zα) 3 …”

Section: Negative Computationsmentioning

confidence: 99%

“…Note that G = x 3 is central and that Z = Z(G) = x 3 × y 3 is abelian of type (3,3). Furthermore, we have xy = yx 4 , and then multiplying by…”

Section: Negative Computationsmentioning

confidence: 99%

“…These are the two nonabelian groups of order 3 3 and at most three groups of order 3 4 . As we will see, the computations of the next section eliminate two of these possibilities, sharpening [ 3 , the direct product of the quaternion group of order 8 with the cyclic group of order 3.…”

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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Section: The Main Resultsmentioning

confidence: 99%

“…Obviously, N is multiplicative and the ordering of the factors in N does not matter. In particular, in Z[G] we have (zα) 3 …”

Section: Negative Computationsmentioning

confidence: 99%

“…Note that G = x 3 is central and that Z = Z(G) = x 3 × y 3 is abelian of type (3,3). Furthermore, we have xy = yx 4 , and then multiplying by…”

Section: Negative Computationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Liu

Passman

2010

J. Algebra Appl.

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“…It also follows directly from Theorem 1 by noting that £ and its algebraic conjugates have bounded absolute values. An analogous result of interest is [HLP,Corollary 2.8].…”

Section: The Semisimple Casementioning

confidence: 90%