فهرست مطالب

International Journal of Group Theory
Volume:4 Issue: 3, Sep2015

  • تاریخ انتشار: 1394/08/22
  • تعداد عناوین: 5
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  • Alessio Russo Pages 1-5
    Let Γ be a normal subgroup of the full automorphism group Aut(G) of a group G, and assume that Inn(G)≤Γ. An endomorphism σ of G is said to be {\it Γ -central} if σ induces the the identity on the factor group G/C G (Γ). Clearly, if Γ=Inn(G), then a Γ -central endomorphism is a {\it central} endomorphism. In this article the conditions under which a Γ -central endomorphism of a group is an automorphism are investigated.
    Keywords: central endomorphism, autocentral endomorphism, purely non, abelian group
  • Alexey Stepanov Pages 7-12
    The present note arises from the author's talk at the conference ``Ischia Group Theory 2014''. For subgroups FleN of a group G denote by Lat(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=Lat(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 Lat(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.
    Keywords: subgroup structure, sandwich classification, Chevalley groups, commutative rings
  • Marco Pellegrini, Maria Tamburini Pages 13-19
    ‎Let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree ≤7 over‎ ‎an algebraically closed field F of characteristic p≥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.
    Keywords: Finite simple groups, $(2, 3)$, generation, Hurwitz generation
  • Gunnar Traustason Pages 21-75
    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 few simplifications and in particular we are able to merge two of the key steps into one.
    Keywords: 4, Engel, nilpotence, left Engel
  • Arye Juhasz Pages 77-177
    Let A be an Artin group with standard generators X={x 1, …,x n}, n≥1 and defining graph Γ A. A \emph{standard parabolic subgroup} of A is a subgroup generated by a subset of X. For elements u and v of A we say (as usual) that u is conjugate to v by an element h of A if h −1 uh=v holds in A. Similarly, if K and L are subsets of A then K is conjugate to L by an element h of A if h −1 Kh=L. In this work we consider the conjugacy of elements and standard parabolic subgroups of a certain type of Artin groups. Results in this direction occur in occur in papers by Duncan, Kazachkov, Remeslennikov, Fenn, Dale, Jun, Godelle, Gonzalez-Meneses, Wiest, Paris, Rolfsen, for example. Of particular interest are centralisers of elements, and of standard parabolic subgroups, normalisers of standard parabolic subgroups and commensurators of parabolic subgroups. In this work we consider similar problems in a new class of Artin groups, introduced in the paper "On relatively extralarge Artin groups and their relative asphericity", by Juhasz, where the word problem is solved, among other things. Also, intersections of parabolic subgroups and their conjugates are considered.
    Keywords: Artin groups, conjugacy, standard parabolic subgroups