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Let $G$ be a finite group and let $π (G) = {p_{1}, p_{2}, ..., p_{k}}$ be the set of prime divisors of $| G |$ for which $p_{1} < p_{2} < \dots < p_{k}$ . The Gruenberg-Kegel graph of $G$ , denoted $GK (G)$ , is defined as follows: its vertex set is $π (G)$ and two different vertices $p_{i}$ and $p_{j}$ are adjacent by an edge if and only if $G$ contains an element of order $p_{i} p_{j}$ . The degree of a vertex $p_{i}$ in $GK (G)$ is denoted by $d_{G} (p_{i})$ and the $k$ -tuple $D (G) = (d_{G} (p_{1}), d_{G} (p_{2}), ..., d_{G} (p_{k}))$ is said to be the degree pattern of $G$ . Moreover, if $ω \subseteq π (G)$ is the vertex set of a connected component of $GK (G)$ , then the largest $ω$ -number which divides $| G |$ , is said to be an...

Some results on the recognizability of the linear groups over the binary field

Mohammad Reza Darafsheh; Yaghoub Farjami; M. Khademi; Ali Reza Moghaddamfar — 2005

Commentationes Mathematicae Universitatis Carolinae

In this paper, we first find the set of orders of all elements in some special linear groups over the binary field. Then, we will prove the characterizability of the special linear group $PSL (13, 2)$ using only the set of its element orders.

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About noncommuting graphs.

Generalized Pascal triangles and Toeplitz matrices.

Recognition of some linear groups over the binary field by the spectrum.

On the noncommuting graph associated with a finite group.

Certain matrices related to Fibonacci sequence having recursive entries.

New families of integer matrices whose leading principal minors form some well-known sequences.

On recognition of the projective special linear groups over binary fields.

Several quantitative characterizations of some specific groups

Some results on the recognizability of the linear groups over the binary field