Results 231 to 240 of about 13,095,517 (269)
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1998
We saw in Chapter 5 that for each n, the set Un of units in ℤn forms a group under multiplication. Our aim in this chapter is to understand more about multiplication and division in ℤn by studying the structure of this group. An important result is that if n = pe, where p is an odd prime, then Un is cyclic; following a commonly-used strategy, we shall ...
Gareth A. Jones, J. Mary Jones
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We saw in Chapter 5 that for each n, the set Un of units in ℤn forms a group under multiplication. Our aim in this chapter is to understand more about multiplication and division in ℤn by studying the structure of this group. An important result is that if n = pe, where p is an odd prime, then Un is cyclic; following a commonly-used strategy, we shall ...
Gareth A. Jones, J. Mary Jones
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Alternating units as free factors in the group of units of integral group rings
Proceedings of the Edinburgh Mathematical Society, 2011AbstractLet G be a group of odd order that contains a non-central element x whose order is either a prime p ≥ 5 or 3l, with l ≥ 2. Then, in $\mathcal{U}(\mathbb{Z}G)$, the group of units of ℤG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large ...
Gonçalves, Jairo Z., Veloso, Paula M.
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2007
Summary: We give a complete characterization of the unit group \(\mathcal U(FS_3)\) of the group algebra \(FS_3\) of the symmetric group \(S_3\) of degree 3 over a finite field \(F\). Moreover, over the prime fields \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\), presentations of the unit groups of the group algebras \(\mathbb{Z}_2S_3\) and \(\mathbb{Z}_3S_3 ...
Sharma, R. K. +2 more
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Summary: We give a complete characterization of the unit group \(\mathcal U(FS_3)\) of the group algebra \(FS_3\) of the symmetric group \(S_3\) of degree 3 over a finite field \(F\). Moreover, over the prime fields \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\), presentations of the unit groups of the group algebras \(\mathbb{Z}_2S_3\) and \(\mathbb{Z}_3S_3 ...
Sharma, R. K. +2 more
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The Ideal Class Group and the Unit Group
1977Recall that the ideal class group of a number ring R consists of equivalence classes of nonzero ideals under the relation $$ I\sim J\quad iff\quad \alpha I = \beta J\quad for{\text{ }}some{\text{ }}non{\text{ }}zero{\text{ }}\alpha ,\beta \in R; $$ the group operation is multiplication defined in the obvious way, and the fact that this is ...
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Units in Group Algebras of Finite p-Groups
Journal of Mathematical Sciences, 2003zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Bovdi, Adalbert, Patay, Zoltán
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1999
For a commutative ring R with identity and an arbitrary group G, let RG denote the group ring of G over R and U(RG) its group of units. It is of interest, see the survey by Dennis (1977), to determine the necessary and sufficient conditions on R and G in order that U(RG) has a specific group-theoretic property, e.g., solvability, nilpotence, etc ...
Ashwani K. Bhandari, I. B. S. Passi
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For a commutative ring R with identity and an arbitrary group G, let RG denote the group ring of G over R and U(RG) its group of units. It is of interest, see the survey by Dennis (1977), to determine the necessary and sufficient conditions on R and G in order that U(RG) has a specific group-theoretic property, e.g., solvability, nilpotence, etc ...
Ashwani K. Bhandari, I. B. S. Passi
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ON THE DERIVED LENGTH OF THE GROUP OF UNITS OF A GROUP ALGEBRA
Journal of Algebra and Its Applications, 2007Let G be a group with cyclic derived subgroup of p-power order (p > 2), and F a field of characteristic p. We determine the derived length of the group of units of the group algebra FG.
Balogh, Zsolt, Li, Yuanlin
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On the derived length of the unit group of a group algebra
Journal of Group Theory, 2010Let \(KG\) be the group algebra of the torsion nilpotent group \(G\) over the field \(K\) of positive charateristic \(p\) with the unit group \(U(KG)\) solvable. Conditions under which the unit group is solvable were investigated by several authors in various cases, but little is known about the derived length of the unit group. \textit{C.
CATINO, Francesco, Ernesto SPINELLI
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Group Identities on Units and Symmetric Units of Group Rings
2010Group Identities on Units of Group Rings.- Group Identities on Symmetric Units.- Lie Identities on Symmetric Elements.- Nilpotence of ...
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The unit group of the modular small group algebra
Mathematical Journal of Okayama University, 1995Let \(G\) be a finite \(p\)-group whose commutator subgroup is elementary Abelian and \(F\) be the field of \(p\) elements. Denote by \(FG\) the group ring and by \(I(G)\) the kernel of the augmentation mapping. Denote by \(G_2\) the commutator subgroup of \(G\).
Salim, Mohamed A.M., Sandling, Robert
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