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Register a new userProof of a conjecture of Abdollahi-Akbari-Maimani concerning the non-commutative graph of finite groups
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The non--commuting graph $Gamma(G)$ of a non--abelian group $G$ is defined as follows. The vertex set $V(Gamma(G))$ of $Gamma(G)$ is $Gsetminus Z(G)$ where $Z(G)$ denotes the center of $G$ and two vertices $x$ and $y$ are adjacent if and only if $xy eq yx$. For non--abelian finite groups $G$ and $H$ it is conjectured that if $Gamma(G) cong Gamma(H)$, then $|G|=|H|$. We prove the conjecture.
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Building on earlier papers of several authors, we establish that there exists a universal constant $c > 0$ such that the minimal base size $b(G)$ of a primitive permutation group $G$ of degree $n$ satisfies $log |G| / log n leq b(G) < 45 (log |G| / log n) + c$. This finishes the proof of Pybers base size conjecture. An ingredient of the proof is that for the distinguishing number $d(G)$ (in the sense of Albertson and Collins) of a transitive permutation group $G$ of degree $n > 1$ we have the estimates $sqrt[n]{|G|} < d(G) leq 48 sqrt[n]{|G|}$.
In a 2004 article, Udo Baumgartner and George Willis used ideas from the structure theory of totally disconnected, locally compact groups to achieve a better understanding of the contraction group U_f associated with an automorphism f of such a group G, assuming that G is metrizable. (Recall that U_f consists of all group elements x such that f^n(x) tends to the identity element as n tends to infinity). Recently, Wojciech Jaworski showed that the main technical tool of the latter article remains valid in the non-metrizable case. He asserted without proof that, therefore, all results from that article remain valid. However, metrizability enters the arguments at a second point. In this note, we resolve this difficulty, by providing an affirmative answer to a question posed by Willis in 2004.
In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.
We prove an elementary lemma concerning primitive amalgams and use it to greatly simplify the proof of the Sims conjecture in the case of almost simple groups.
We exhibit a regular language of geodesics for a large set of elements of $BS(1,n)$ and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of $BS(1,n)$, which was initially computed by Collins, Edjvet and Gill in [5]. Our methods are based on those we develop in [8] to show that $BS(1,n)$ has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan, Duchin and Kropholler in [1].
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