فهرست مطالب
International Journal of Group Theory
Volume:8 Issue: 3, Sep 2019
- تاریخ انتشار: 1398/06/10
- تعداد عناوین: 5
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Pages 1-8Let $G$ be a finite group. In this paper, we study the structure of finite groups having $|G|-r$ cyclic subgroups for $3leq rleq 5$.Keywords: Finite groups, cyclic subgroups, $p$-groups
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Pages 9-14Let $Gamma$ be the first Grigorchuk group. According to a result of Bar-thol-di, the only left Engel elements of $Gamma$ are the involutions. This implies that the set of left Engel elements of $Gamma$ is not a subgroup. The natural question arises whether this is also the case for the sets of bounded left Engel elements, right Engel elements and bounded right Engel elements of $Gamma$. Motivated by this, we prove that these three subsets of $Gamma$ coincide with the identity subgroup.Keywords: Engel elements, Grigorchuk group
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Pages 15-31A finite group $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ of elements in $G$ with the property that for any $k$ nontrivial elements $s_1, s_2,ldots,s_k$ in $G$ there exists $yin C$ such that $G = langle s_i,yrangle$ for $i=1, 2,ldots,k$. Further, the exact uniform spread of $G$ is the largest $k$ such that $G$ has the uniform spread $k$. In this paper we give upper bounds on the exact uniform spreads of thirteen sporadic simple groups.Keywords: Exact uniform spread, Exact spread, Sporadic group
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Pages 33-42Let $G_{Gamma}$ be a partially commutative group. We find a finite presentation for the subgroup $Conjv(G_{Gamma})$ of compressed vertex conjugating automorphisms of the automorphism group $Aut(G_{Gamma})$ of $G_{G}$. We have written GAP packages which compute presentations for $Aut(G_{Gamma})$ and its subgroups $Conj(G_{Gamma})$ and $Conjv(G_{Gamma})$.Keywords: Partially commutative groups, Right-angled Artin groups, Automorphisms of groups
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Pages 43-64In this paper we construct transitive $t$-designs from the linear groups $L(2,q), q leq 23$.
Thereby we classify $t$-designs, $t ge 2$, admitting a transitive action of the linear groups $L(2,q), q leq 23$, up to 35 points and obtained numerous transitive designs, for $36leq vleq 55$. In many cases we proved the existence of $t$-designs with certain parameter sets. Among others we constructed $t$-designs with parameters $2$-$(55,10,4)$, $3$-$(24,11,495)$, $3$-$(24,12, 5m), m in {11, 12,22, 33, 44, 66, 132}$. Furthermore, we constructed strongly regular graphs admitting a transitive action of the linear groups $L(2,q), q leq 23$.Keywords: $t$-design, strongly regular graph, linear group, Transitive group