Permutable altgruplu bir grubun çözülebilirliği üzerine
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Date
2016
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İki çözülebilir grubun çarpımının çözülebililir olmayabileceği bilinmektedir. Bu tezde, V. S. Monakhov'un makalesine dayanarak $G=AB$ tipindeki sonlu grubun çözülebilirliği çalışılmıştır. Bir $G$ grubunun nilpotent ve öz-normalleyen bir altgrubu var ise, bu altgruba $G$'nin Carter altgrubu denir. $G$ grubunun süperçözülebilir bir $H$ altgrubuna $H \leq H_1 < T \leq G$ iken $|T:H_1|$ asal değildir koşulunu sağlıyor ise $G$'nin Gashutz altgrubu denir. Monakhov, Kegel-Weiland ve Kazarin'nin sonuçlarını kullanarak gösteriyor ki eğer $A$'nın her Carter altgrubu, $B$'nin her Carter altgrubu ile degişmeli ise $G=AB$ çözülebilirdir. Ayrica $G=AB$'nin çözülebilirliğini $A$'nın her Carter altgrubunun tekil mertebeli ve $B$'nin her Gashutz altgrubu ile değişmeli olmasi koşulu altında da vermektedir. Bunun yanı sıra, okuyucuya kolaylık sağlaması için tezde kullanılan Carter altgruplarının özellikleri Roger W. Carter'ın ``On nilpotent self-normalizing subgroups of soluble groups'' adlı makalesinden ispatları açıklanarak verilmiştir.
It is well-known that a product of two solvable groups need not to be solvable. In this thesis, depending on an article of V. S. Monakhov [10], the solvability of a finite group $G=AB$ is studied. A subgroup $K$ of a group $G$ is called a Carter subgroup if $K$ is nilpotent and self-normalizing. A supersolvable subgroup $H$ of a group $G$ is called a Gaschütz subgroup if the condition $H\leq H_1< T\leq G$ implies that $\left|T:H_1\right|$ is not prime. Using the Kegel-Wielandt and Kazarin results, Monakhov showed that if every Carter subgroup of $A$ commutes with every Carter subgroup of $B$, then $G=AB$ is solvable. Moreover, he gives that $G=AB$ is solvable when every Carter subgroup of $A$ is of odd order and commutes with every Gaschütz subgroup of $B$. In addition, for convenience of the reader, the proofs of the properties of Carter subgroups given in the article ``On nilpotent self-normalizing subgroups of solvable groups' of Roger.~W.~Carter are clarified.
It is well-known that a product of two solvable groups need not to be solvable. In this thesis, depending on an article of V. S. Monakhov [10], the solvability of a finite group $G=AB$ is studied. A subgroup $K$ of a group $G$ is called a Carter subgroup if $K$ is nilpotent and self-normalizing. A supersolvable subgroup $H$ of a group $G$ is called a Gaschütz subgroup if the condition $H\leq H_1< T\leq G$ implies that $\left|T:H_1\right|$ is not prime. Using the Kegel-Wielandt and Kazarin results, Monakhov showed that if every Carter subgroup of $A$ commutes with every Carter subgroup of $B$, then $G=AB$ is solvable. Moreover, he gives that $G=AB$ is solvable when every Carter subgroup of $A$ is of odd order and commutes with every Gaschütz subgroup of $B$. In addition, for convenience of the reader, the proofs of the properties of Carter subgroups given in the article ``On nilpotent self-normalizing subgroups of solvable groups' of Roger.~W.~Carter are clarified.
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Matematik, Mathematics
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46