On central endomorphisms of a group
Alessio
Russo
Seconda Universita di Napoli
author
text
article
2015
eng
Let $\Gamma$ be a normal subgroup of the full automorphism group $Aut(G)$ of a group $G$, and assume that $Inn(G)\leq \Gamma$. An endomorphism $\sigma$ of $G$ is said to be $\Gamma$-central if $\sigma$ induces the the identity on the factor group $G/C_G(\Gamma)$. Clearly, if $\Gamma=Inn(G)$, then a $\Gamma$-central endomorphism is a central endomorphism. In this article the conditions under which a $\Gamma$-central endomorphism of a group is an automorphism are investigated.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
3
no.
2015
1
5
http://ijgt.ui.ac.ir/article_9970_ca5566bf28c2bba5f0b9d963287fbb1c.pdf
dx.doi.org/10.22108/ijgt.2015.9970
Sandwich classification theorem
Alexey
Stepanov
St.Petersburg State University
author
text
article
2015
eng
The present note arises from the author's talk at the conference ``Ischia Group Theory 2014''. For subgroups $F\le N$ of a group $G$ denote by $L(F,N)$ the set of all subgroups of $N$, containing $F$. Let $D$ be a subgroup of $G$. In this note we study the lattice $LL=L(D,G)$ and the lattice $LL'$ of subgroups of $G$, normalized by $D$. We say that $LL$ satisfies sandwich classification theorem if $LL$ splits into a disjoint union of sandwiches $L(F,N_G(F))$ over all subgroups $F$ such that the normal closure of $D$ in $F$ coincides with $F$. Here $N_G(F)$ denotes the normalizer of $F$ in $G$. A similar notion of sandwich classification is introduced for the lattice $LL'$. If $D$ is perfect, i.\,e. coincides with its commutator subgroup, then it turns out that sandwich classification theorem for $LL$ and $LL'$ are equivalent. We also show how to find basic subroup $F$ of sandwiches for $LL'$ and review sandwich classification theorems in algebraic groups over rings.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
3
no.
2015
7
12
http://ijgt.ui.ac.ir/article_10515_09350789c2e261ccde151cedd39968fe.pdf
dx.doi.org/10.22108/ijgt.2015.10515
Finite simple groups of low rank: Hurwitz generation and $(2,3)$-generation
Marco
Pellegrini
Universita Cattolica del Sacro Cuore
author
Maria
Tamburini
Universita Cattolica del Sacro Cuore
author
text
article
2015
eng
Let us consider the set of non-abelian finite simple groups which admit non-trivial irreducible projective representations of degree $\le 7$ over an algebraically closed field $F$ of characteristic $p\geq 0$. We survey some recent results which lead to the complete list of the groups in this set which are $(2, 3, 7)$-generated and of those which are $(2,3)$-generated.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
3
no.
2015
13
19
http://ijgt.ui.ac.ir/article_10630_d76e418244e79725ce81061e94048b61.pdf
dx.doi.org/10.22108/ijgt.2015.10630
The local nilpotence theorem for 4-Engel groups revisited
Gunnar
Traustason
University of Bath
author
text
article
2015
eng
The proof of the local nilpotence theorem for $4$-Engel groups was completed by G. Havas and M. Vaughan-Lee in 2005. The complete proof on the other hand is spread over several articles and the aim of this paper is to give a complete coherent linear version. In the process we are also able to make a few simplifications and in particular we are able to merge two of the key steps into one.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
3
no.
2015
21
75
http://ijgt.ui.ac.ir/article_5551_c975d089a43eec9ab751688d5f9df967.pdf
dx.doi.org/10.22108/ijgt.2015.5551
Conjugacy in relatively extra-large Artin groups
Arye
Juhasz
Italy
author
text
article
2015
eng
In this work we consider conjugacy of elements and parabolic subgroups in details, in a new class of Artin groups, introduced in an earlier work, which may contain arbitrary parabolic subgroups. In particular, we find algorithmically minimal representatives of elements in a conjugacy class and also an algorithm to pass from one minimal representative to the others.
International Journal of Group Theory
University of Isfahan
2251-7650
4
v.
3
no.
2015
77
117
http://ijgt.ui.ac.ir/article_11019_6eed0378df31f689d018e18d852b49d7.pdf
dx.doi.org/10.22108/ijgt.2015.11019