Flag-transitive block designs with prime replication number and almost simple groups

Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

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CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

International Journal of Algebra and Computation ◽

10.1142/s021819671000587x ◽

2010 ◽

Vol 20 (07) ◽

pp. 847-873 ◽

Cited By ~ 11

Author(s):

Z. AKHLAGHI ◽

B. KHOSRAVI ◽

M. KHATAMI

Keyword(s):

Finite Group ◽

Group Theory ◽

Prime Number ◽

Prime Graph ◽

Composition Factor ◽

Simple Groups ◽

Almost Simple Groups ◽

Prime Graphs ◽

A Finite Group ◽

Nonabelian Composition Factor

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

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3-Signalizers in almost simple groups

Finite Groups 2003 ◽

10.1515/9783110198126.229 ◽

2004 ◽

pp. 229-246

Keyword(s):

Simple Groups ◽

Almost Simple Groups

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Number of Sylow subgroups in finite groups

Journal of Group Theory ◽

10.1515/jgth-2018-0010 ◽

2018 ◽

Vol 21 (4) ◽

pp. 695-712 ◽

Cited By ~ 1

Author(s):

Wenbin Guo ◽

Evgeny P. Vdovin

Keyword(s):

Finite Groups ◽

Simple Groups ◽

Alternative Proof ◽

Sylow Subgroups ◽

Almost Simple Groups

AbstractDenote by {\nu_{p}(G)} the number of Sylow p-subgroups of G. It is not difficult to see that {\nu_{p}(H)\leqslant\nu_{p}(G)} for {H\leqslant G}, however {\nu_{p}(H)} does not divide {\nu_{p}(G)} in general. In this paper we reduce the question whether {\nu_{p}(H)} divides {\nu_{p}(G)} for every {H\leqslant G} to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof of Navarro’s theorem.

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ON A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972719001060 ◽

2019 ◽

Vol 102 (1) ◽

pp. 77-90

Author(s):

PABLO SPIGA

Keyword(s):

Fixed Point ◽

Finite Group ◽

Simple Group ◽

Simple Groups ◽

Alternating Group ◽

Permutation Groups ◽

Sporadic Simple Group ◽

Almost Simple Groups ◽

A Finite Group ◽

Groups Of Lie Type

Let $G$ be a finite group with two primitive permutation representations on the sets $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ and let $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{2}$, that is, for every $g\in G$, $\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if $\unicode[STIX]{x1D70B}_{2}(g)=0$. We have conjectured in Spiga [‘Permutation characters and fixed-point-free elements in permutation groups’, J. Algebra299(1) (2006), 1–7] that under this hypothesis either $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}_{2}$ or one of $\unicode[STIX]{x1D70B}_{1}-\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{2}-\unicode[STIX]{x1D70B}_{1}$ is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of $G$ is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.

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