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If ℱ is a class of groups, then a minimal non-ℱ-group (a dual minimal non-ℱ-group resp.) is a group which is not in ℱ but any of its proper subgroups (factor groups resp.) is in ℱ. In many problems of classification of groups it is sometimes useful to know structure properties of classes of minimal non-ℱ-groups and dual minimal non-ℱ-groups. In fact, the literature on group theory contains many results directed to classify some of the most remarkable among the aforesaid classes. In particular, V....

On recognition of finite simple groups with connected prime graph.

Vasil'ev, A.V., Gorshkov, I.B. (2009)

Sibirskij Matematicheskij Zhurnal

On self-centralizing Sylow subgroups of order four

Rolf Brandl, Valeria Fedri, Luigi Serena (1996)

Rendiconti del Seminario Matematico della Università di Padova

On the composition factors of a group with the same prime graph as $B_{n} (5)$

Azam Babai, Behrooz Khosravi (2012)

Czechoslovak Mathematical Journal

Let $G$ be a finite group. The prime graph of $G$ is a graph whose vertex set is the set of prime divisors of $| G |$ and two distinct primes $p$ and $q$ are joined by an edge, whenever $G$ contains an element of order $p q$ . The prime graph of $G$ is denoted by $Γ (G)$ . It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if $G$ is a finite group such that $Γ (G) = Γ (B_{n} (5))$ , where $n \geq 6$ , then $G$ has a unique nonabelian composition factor isomorphic to $B_{n} (5)$ or $C_{n} (5)$ .

On the computation of the order of Janko’s first simple group $𝔉_{1}$

Herbert Dietzsch (1971)

Rendiconti del Seminario Matematico della Università di Padova

On the finite simple groups

Zvonimir Janko (1976/1977)

Séminaire Bourbaki

On the recognizability of some projective general linear groups by the prime graph

Masoumeh Sajjadi (2022)

Commentationes Mathematicae Universitatis Carolinae

Let $G$ be a finite group. The prime graph of $G$ is a simple graph $Γ (G)$ whose vertex set is $π (G)$ and two distinct vertices $p$ and $q$ are joined by an edge if and only if $G$ has an element of order $p q$ . A group $G$ is called $k$ -recognizable by prime graph if there exist exactly $k$ nonisomorphic groups $H$ satisfying the condition $Γ (G) = Γ (H)$ . A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that $PGL (2, p^{α})$ is recognizable, if $p$ is an odd prime and $α > 1$ is odd. But for even $α$ , only the recognizability...