International Journal of Group Theory
Volume:8 Issue: 3, Sep 2019

Let $G$ be a finite group. In this paper, we study the structure of finite groups having $|G|-r$ cyclic subgroups for $3leq rleq 5$.

Let $Gamma$ 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.

‎‎A 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.

‎Let $G_{Gamma}$ 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 $Conjv(G_{Gamma})$‎.

‎In 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 $L(2,q)‎, ‎q leq 23$‎.