فهرست مطالب
International Journal of Group Theory
Volume:12 Issue: 4, Dec 2023
- تاریخ انتشار: 1401/11/25
- تعداد عناوین: 5
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Pages 223-226In this note, we prove that if $G$ is solvable and ${\rm cod}(\chi)$ is a $p$-power for every nonlinear, monomial, monolithic $\chi\in {\rm Irr}(G)$ or every nonlinear, monomial, monolithic $\chi \in {\rm IBr} (G)$, then $P$ is normal in $G$, where $p$ is a prime and $P$ is a Sylow $p$-subgroup of $G$.Keywords: monomial Characters, monolithic characters, character codegrees
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Pages 227-235The classification of trivectors(trilinear alternating forms) depends essentially on the dimension $n$ of the base space. This classification seems to be a difficult problem (unlike in the bilinear case). For $n\leq 8 $ there exist finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\bar{K}$(the algebraic closure of $K$) to $K.$ In this paper, we are interested in the classification of trivectors of an eight dimensional vector space over a finite field of characteristic $3,$ $% K=\mathbb{F}_{3^{m}}.$ We obtain a $31$ inequivalent trivectors, $20$ of which are full rank. Having its motivation in the theory of the generalized elliptic curves and commutative moufang loop, this research studies the case of the forms over the 3 elements field. We use a transfer theorem providing a one-to-one correspondence between the classes of trilinear alternating forms of rank $8$ over a finite field with $3$ elements $\mathbb{F}_{3}$ and the rank $9$ class $2$ Hall generalized elliptic curves (GECs) of $3$-order $9$ and commutative moufang loop (CMLs). We derive a classification and explicit descriptions of the $31$ Hall GECs whose rank and $3$-order both equal $9$ and the number of order $3^{9}$-CMLs.Keywords: Commutative moufang loops, Generalized elliptic curves, Trivectors, Classification
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Pages 237-252Groups associated to surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. Gül and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit ramification structures. We extend their result by showing that quotients of generalisations of the GGS-groups, namely multi-EGS groups, also admit ramification structures.Keywords: Groups acting on rooted trees, finite p-groups, ramification structures
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Pages 253-264We provide some characterization theorems about just infinite profinite residually solvable Lie algebras, similarly to what C. Reid has done for just infinite profinite groups. In particular, we prove that a profinite residually solvable Lie algebra is just infinite if and only if its obliquity subalgebra has finite codimension in the Lie algebra, and we establish a criterion for a profinite residually solvable Lie algebra to be just infinite, looking at the finite Lie algebras occurring in the inverse system.Keywords: just-infinite Lie algebras, profinite Lie algebras, residually solvable Lie algebras
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Pages 265-269Olga Taussky-Todd suggested the problem of determining the possible values of integer circulant determinants. To solve a special case of the problem, Laquer gave a factorization of circulant determinants. In this paper, we give a modest generalization of Laquer's theorem. Also, we give an application of the generalization to integer group determinants.Keywords: Integer circulant determinant, Integer group determinant, Circulant determinant, group determinant, Dedekind' s theorem