## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 45, Number 3 (2016), 337-363.

### A New Characterization of Some Simple Groups by Order and Degree Pattern of Solvable Graph

B. AKBARI, N. IIYORI, and A. R. MOGHADDAMFAR

#### Abstract

The solvable graph of a finite group $G$, denoted by ${\Gamma}_{\rm s}(G)$, is a simple graph whose vertices are the prime divisors of $|G|$ and two distinct primes $p$ and $q$ are joined by an edge if and only if there exists a solvable subgroup of $G$ such that its order is divisible by $pq$. Let $p_1<p_2<\cdots<p_k$ be all prime divisors of $|G|$ and let ${\rm D}_{\rm s}(G)=(d_{\rm s}(p_1), d_{\rm s}(p_2), \ldots, d_{\rm s}(p_k))$, where $d_{\rm s}(p)$ signifies the degree of the vertex $p$ in ${\Gamma}_{\rm s}(G)$. We will simply call ${\rm D}_{\rm s}(G)$ the degree pattern of solvable graph of $G$. In this paper, we determine the structure of any finite group $G$ (up to isomorphism) for which ${\Gamma}_{\rm s}(G)$ is star or bipartite. It is also shown that the sporadic simple groups and some of projective special linear groups $L_2(q)$ are characterized via order and degree pattern of solvable graph.

#### Article information

**Source**

Hokkaido Math. J., Volume 45, Number 3 (2016), 337-363.

**Dates**

First available in Project Euclid: 7 November 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1478487614

**Digital Object Identifier**

doi:10.14492/hokmj/1478487614

**Mathematical Reviews number (MathSciNet)**

MR3568632

**Zentralblatt MATH identifier**

06653107

**Subjects**

Primary: 20D05: Finite simple groups and their classification 20D06: Simple groups: alternating groups and groups of Lie type [See also 20Gxx] 20D08: Simple groups: sporadic groups 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17] 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]

**Keywords**

solvable graph degree pattern simple group $OD_s$-characterization of a finite group

#### Citation

AKBARI, B.; IIYORI, N.; MOGHADDAMFAR, A. R. A New Characterization of Some Simple Groups by Order and Degree Pattern of Solvable Graph. Hokkaido Math. J. 45 (2016), no. 3, 337--363. doi:10.14492/hokmj/1478487614. https://projecteuclid.org/euclid.hokmj/1478487614