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1. CMB 2010 (vol 54 pp. 237)
The Structure of the Unit Group of the Group Algebra ${\mathbb{F}}_{2^k}D_{8}$
Let $RG$ denote the group ring of the group $G$ over
the ring $R$. Using an isomorphism between $RG$ and a
certain ring of $n \times n$ matrices in conjunction with other
techniques, the structure of the unit group of the group algebra
of the dihedral group of order $8$ over any
finite field of chracteristic $2$ is determined in
terms of split extensions of cyclic groups.
Categories:16U60, 16S34, 20C05, 15A33 |
2. CMB 2005 (vol 48 pp. 80)
Trivial Units for Group Rings with $G$-adapted Coefficient Rings For each finite group $G$ for which the integral group ring
$\mathbb{Z}G$ has only trivial units, we give ring-theoretic
conditions for a commutative ring $R$ under which the group ring
$RG$ has nontrivial units. Several examples of rings satisfying
the conditions and rings not satisfying the conditions are given.
In addition, we extend a well-known result for fields by showing
that if $R$ is a ring of finite characteristic and $RG$ has only
trivial units, then $G$ has order at most 3.
Categories:16S34, 16U60, 20C05 |
3. CMB 2001 (vol 44 pp. 27)
Normal Subloops in the Integral Loop Ring of an $\RA$ Loop We show that an $\RA$ loop has a torsion-free normal complement in the
loop of normalized units of its integral loop ring. We also
investigate whether an $\RA$ loop can be normal in its unit loop.
Over fields, this can never happen.
Categories:20N05, 17D05, 16S34, 16U60 |
4. CMB 2000 (vol 43 pp. 60)
Trivial Units in Group Rings Let $G$ be an arbitrary group and let $U$ be a subgroup of the
normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$
as a subgroup of finite index, then $U = G$. This result can be used
to give an alternative proof of a recent result of Marciniak and
Sehgal on units in the integral group ring of a crystallographic group.
Keywords:units, trace, finite conjugate subgroup Categories:16S34, 16U60 |
5. CMB 1997 (vol 40 pp. 103)
The transfer of a commutator law from a nil-ring to its adjoint group For every field $F$ of characteristic $p\geq 0$,
we construct an example of a finite dimensional nilpotent
$F$-algebra $R$ whose adjoint group $A(R)$ is not
centre-by-metabelian, in spite of the fact that $R$ is Lie
centre-by-metabelian
and satisfies the identities $x^{2p}=0$ when $p>2$ and
$x^8=0$ when $p=2$. The
existence of such algebras answers a question raised by
A.~E.~Zalesskii, and is in contrast to
positive results obtained by Krasilnikov, Sharma and Srivastava
for Lie metabelian rings
and by Smirnov for the class Lie centre-by-metabelian nil-algebras
of exponent 4 over a field of characteristic 2 of cardinality at least 4.
Categories:16U60, 17B60 |