International Journal of Group Theory
Volume:9 Issue: 3, Sep 2020

‎Let $G$ be a finite group in which every Sylow subgroup‎ ‎is seminormal or abnormal‎. ‎We prove that $G$ has a Sylow tower‎. ‎We establish that if a group has a maximal subgroup ‎‎‎‎with Sylow subgroups under the same conditions‎, ‎then this group is soluble‎.

‎Groups of $F$-type were introduced in [B‎. ‎Fine and G‎. ‎Rosenberger‎, ‎Generalizing Algebraic Properties of Fuchsian Groups‎, emph{London Math. Soc. Lecture Note Ser.}, textbf{159} (1991)‎ ‎124--147.] as a natural algebraic generalization of Fuchsian groups‎. ‎They can be considered as the analogs of cyclically pinched one-relator groups where torsion is allowed‎. ‎Using the methods In [B‎. ‎Fine‎. ‎M‎. ‎Kreuzer and G‎. ‎Rosenberger‎, ‎Faithful Real Representations of Cyclically Pinched One-Relator Groups, Int. J. Group Theory, 3 ‎(2014) 1--8.] we prove that any hyperbolic group of $F$-type has a faithful representation in $PSL(2,mathbb R)$‎. ‎From this we also obtain that a cyclically pinched one-relator group has a faithful real representation if and only if it is hyperbolic‎. ‎We further survey the many nice properties of groups of $F$-type‎. ‎

We determine the groups of minimal order in which all groups of order $n$ can embedded for $ 1 le n le 15$‎. ‎We further determine the minimal order of a group in which all groups of order $n$ or less can be embedded‎, ‎also for $ 1 le n le 15$‎.

In this note we show that for any powerful $p$-group $G$‎, ‎the subgroup‎ ‎$Omega_{i}(G^{p^{j}})$ is powerfully nilpotent for all $i,jgeq1$‎ ‎when $p$ is an odd prime‎, ‎and $igeq1$‎, ‎$jgeq2$ when $p=2$‎. ‎We‎ ‎provide an example to show why this modification is needed in the‎ ‎case $p=2$‎. ‎Furthermore we obtain a bound on the powerful nilpotency‎ ‎class of $Omega_{i}(G^{p^{j}})$‎.

Let $G$ be a finite group and $cd(G)$ denote the character degree set for $G$. The prime graph $DG$ is a simple graph whose vertex set consists of prime divisors of elements in $cd(G)$, denoted $rho(G)$. Two primes $p,qin rho(G)$ are adjacent in $DG$ if and only if $pq|a$ for some $ain cd(G)$. We determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group.