Expand all Collapse all | Results 1 - 10 of 10 |
1. CMB 2011 (vol 56 pp. 344)
Involutions and Anticommutativity in Group Rings Let $g\mapsto g^*$ denote an involution on a
group $G$. For any (commutative, associative) ring
$R$ (with $1$), $*$ extends linearly to an involution
of the group ring $RG$. An element $\alpha\in RG$
is symmetric if $\alpha^*=\alpha$ and
skew-symmetric if $\alpha^*=-\alpha$.
The skew-symmetric elements are closed under
the Lie bracket, $[\alpha,\beta]=\alpha\beta-\beta\alpha$.
In this paper, we investigate when this set is also closed
under the ring product in $RG$.
The symmetric elements are closed under the Jordan
product, $\alpha\circ\beta=\alpha\beta+\beta\alpha$.
Here, we determine when this product is trivial.
These two problems
are analogues of problems about the skew-symmetric and
symmetric elements in group rings that have received a
lot of attention.
Categories:16W10, 16S34 |
2. 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 |
3. CMB 2008 (vol 51 pp. 291)
Group Algebras with Minimal Strong Lie Derived Length Let $KG$ be a non-commutative strongly Lie solvable group algebra of a
group $G$ over a field $K$ of positive characteristic $p$. In this
note we state necessary and sufficient conditions so that the
strong Lie derived length of $KG$ assumes its minimal value, namely
$\lceil \log_{2}(p+1)\rceil $.
Keywords:group algebras, strong Lie derived length Categories:16S34, 17B30 |
4. 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 |
5. CMB 2003 (vol 46 pp. 14)
Generalized Commutativity in Group Algebras We study group algebras $FG$ which can be graded by a finite abelian
group $\Gamma$ such that $FG$ is $\beta$-commutative for a
skew-symmetric bicharacter $\beta$ on $\Gamma$ with values in $F^*$.
Categories:16S34, 16R50, 16U80, 16W10, 16W55 |
6. 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 |
7. CMB 2000 (vol 43 pp. 100)
A Gorenstein Ring with Larger Dilworth Number than Sperner Number A counterexample is given to a conjecture of Ikeda by finding a class of
Gorenstein rings of embedding dimension $3$ with larger Dilworth number than
Sperner number. The Dilworth number of $A[Z/pZ\oplus Z/pZ]$ is computed
when $A$ is an unramified principal Artin local ring.
Categories:13E15, 16S34 |
8. 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 |
9. CMB 1998 (vol 41 pp. 481)
The periodic radical of group rings and incidence algebras Let $R$ be a ring with $1$ and $P(R)$ the periodic radical of $R$.
We obtain necessary and sufficient conditions for $P(\RG) = 0$ when
$\RG$ is the group ring of an $\FC$ group $G$ and $R$ is commutative. We
also obtain a complete description of $P\bigl(I (X, R)\bigr)$ when
$I(X,R)$ is the incidence algebra of a locally finite partially
ordered set $X$ and $R$ is commutative.
Categories:16S34, 16S99, 16N99 |
10. CMB 1998 (vol 41 pp. 109)
On generalized third dimension subgroups Let $G$ be any group, and $H$ be a normal subgroup of $G$. Then M.~Hartl
identified the subgroup $G \cap(1+\triangle^3(G)+\triangle(G)\triangle(H))$
of $G$. In this note we give an independent proof of the result of Hartl,
and we identify two subgroups
$G\cap(1+\triangle(H)\triangle(G)\triangle(H)+\triangle([H,G])\triangle(H))$,
$G\cap(1+\triangle^2(G)\triangle(H)+\triangle(K)\triangle(H))$ of $G$ for
some subgroup $K$ of $G$ containing $[H,G]$.
Categories:20C07, 16S34 |