فهرست مطالب

International Journal of Group Theory - Volume:8 Issue:4, 2019
  • Volume:8 Issue:4, 2019
  • تاریخ انتشار: 1398/09/10
  • تعداد عناوین: 5
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  • Julian Brough * Pages 1-3
    We say that a group $G$ satisfies the one-prime power hypothesis for conjugacy classes if the greatest common divisor for all pairs of distinct conjugacy class sizes are prime powers‎. ‎Insoluble groups which satisfy the one-prime power hypothesis have been classified‎. ‎However it has remained an open question whether the one-prime power hypothesis is inherited by normal subgroups and quotients groups‎. ‎In this note we construct examples to show the one-prime power hypothesis is not necessarily inherited by normal subgroups or quotient groups‎.
    Keywords: ‎Conjugacy classes‎, ‎finite groups‎, ‎restriction to substructures
  • Matthew Conder, Alastair Litterick * Pages 5-9
    We establish the existence of two rigid triples of conjugacy classes in the algebraic group G2 in characteristic 5, complementing results of the second author with Liebeck and Marion. As a corollary, the finite groups G2(5^n) are not (2,4,5)-generated, confirming a conjecture of Marion in this case.
    Keywords: ‎triangle groups‎, ‎finite groups of Lie type‎, ‎representation varieties
  • Michael Vaughan Lee * Pages 11-28
    Graham Higman published two important papers in 1960‎. ‎In the first of these‎ ‎papers he proved that for any positive integer $n$ the number of groups of‎ ‎order $p^{n}$ is bounded by a polynomial in $p$‎, ‎and he formulated his famous‎ ‎PORC conjecture about the form of the function $f(p^{n})$ giving the number of‎ ‎groups of order $p^{n}$‎. ‎In the second of these two papers he proved that the‎ ‎function giving the number of $p$-class two groups of order $p^{n}$ is PORC‎. ‎He established this result as a corollary to a very general result about‎ ‎vector spaces acted on by the general linear group‎. ‎This theorem takes over a‎ ‎page to state‎, ‎and is so general that it is hard to see what is going on‎. ‎Higman's proof of this general theorem contains several new ideas and is quite‎ ‎hard to follow‎. ‎However in the last few years several authors have developed‎ ‎and implemented algorithms for computing Higman's PORC formulae in‎ ‎special cases of his general theorem‎. ‎These algorithms give perspective on‎ ‎what are the key points in Higman's proof‎, ‎and also simplify parts of the proof‎. ‎In this note I give a proof of Higman's general theorem written in the light‎ ‎of these recent developments‎.
    Keywords: ‎PORC, $p$-group, enumerate
  • Reza Hobbi, Mohammad Shahryari * Pages 29-36
    ‎In the space of marked group‎s, ‎we determine the structure of groups which are limit points of the set of all generalized quaternion groups‎.
    Keywords: ‎the space of marked groups‎, ‎Gromov-Grigorchuk metric‎, ‎generalized quaternion groups‎, ‎universal theory‎, ‎ultra-product
  • Mark Bissler, Jacob Laubacher * Pages 37-46
    ‎We investigate prime character degree graphs of solvable groups‎. ‎In particular‎, ‎we consider a family of graphs $Gamma_{k,t}$ constructed by adjoining edges between two complete graphs in a one-to-one fashion‎. ‎In this paper we determine completely which graphs $Gamma_{k,t}$ occur as the prime character degree graph of a solvable group‎.
    Keywords: ‎Character degree graphs‎, ‎solvable groups‎, ‎families