Search results for "Quaternion group"

On the loewy series of the group algebra of groups of smallp-Length∗

1989

(1989). On the loewy series of the group algebra of groups of small p-Length. Communications in Algebra: Vol. 17, No. 5, pp. 1249-1274.

A presentation and a representation of the Held group

1996

In this note we give a brief description of a new presentation of the Held group, which is deduced only from the original work of D. Held in 1969, who shows that a finite simple group, having the same centralizer of a 2-central involution as in the Mathieu group M24, is M24, L5(2) or a group of order 4.030.387.200. The first complete uniqueness proof for the latter case was given by L. Soicher in 1991. The generators and relations occurring here are easy to verify by a simple Todd–Coxeter algorithm. It is an easy task to get a new uniqueness and existence proof of the Held group from this result. Also basic facts like the Schur Multiplier or the automorphism group of the Held group follow f…

Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems

2021

In the present paper we show that it is possible to obtain the well known Pauli group $P=\langle X,Y,Z \ | \ X^2=Y^2=Z^2=1, (YZ)^4=(ZX)^4=(XY)^4=1 \rangle $ of order $16$ as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere $S^3$. The first of these spaces of orbits is realized via an action of the quaternion group $Q_8$ on $S^3$; the second one via an action of the cyclic group of order four $\mathbb{Z}(4)$ on $S^3$. We deduce a result of decomposition of $P$ of topological nature and then we find, in connection with the theory of pseudo-fermions, a possible physical interpretation of this decomposition.

Two groups with isomorphic group algebras

1990

Central Units, Class Sums and Characters of the Symmetric Group

2010

In the search for central units of a group algebra, we look at the class sums of the group algebra of the symmetric group S n in characteristic zero, and we show that they are units in very special instances.

Group algebras whose units satisfy a group identity

1997

Let F G FG be the group algebra of a torsion group over an infinite field F F . Let U U be the group of units of F G FG . We prove that if U U satisfies a group identity, then F G FG satisfies a polynomial identity. This confirms a conjecture of Brian Hartley.

Divisible Designs Admitting, as an Automorphism Group, an Orthogonal Group or a Unitary Group

2001

We construct some divisible designs starting from a projective space. These divisible designs admit an orthogonal group or a unitary group as an automorphism group.

Symmetric units and group identities

1998

In this paper we study rings R with an involution whose symmetric units satisfy a group identity. An important example is given by FG, the group algebra of a group G over a field F; in fact FG has a natural involution induced by setting g?g −1 for all group elements g∈G. In case of group algebras if F is infinite, charF≠ 2 and G is a torsion group we give a characterization by proving the following: the symmetric units satisfy a group identity if and only if either the group of units satisfies a group identity (and a characterization is known in this case) or char F=p >0 and 1) FG satisfies a polynomial identity, 2) the p-elements of G form a (normal) subgroup P of G and G/P is a Hamiltonia…

Symmetric (79, 27, 9)-designs Admitting a Faithful Action of a Frobenius Group of Order 39

1997

AbstractIn this paper we present the classification of symmetric designs with parameters (79, 27, 9) on which a non-abelian group of order 39 acts faithfully. In particular, we show that such a group acts semi-standardly with 7 orbits. Using the method of tactical decompositions, we are able to construct exactly 1320 non-isomorphic designs. The orders of the full automorphism groups of these designs all divide 8 · 3 · 13.

Characterization of strong chain geometries by their automorphism group

1992

A wide class of chain geometries is characterized by their automorphism group using properties of a distinguished involution.