Research Article

Year 2018,
Volume: 24 Issue: 24, 73 - 90, 05.07.2018
### Abstract

### References

- S. Abe, A characterization of some nite simple groups by orders of their solvable subgroups, Hokkaido Math. J., 31(2) (2002), 349-361.
- S. Abe and N. Iiyori, A generalization of prime graphs of nite groups, Hokkaido Math. J., 29(2) (2000), 391-407.
- B. Akbari, N. Iiyori and A. R. Moghaddamfar, A new characterization of some simple groups by order and degree pattern of solvable graph, Hokkaido Math. J., 45(3) (2016), 337-363.
- J. N. Bray, D. F. Holt and C. M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Mathematical Society Lecture Note Series, 407, Cambridge University Press, Cambridge, 2013.
- A. A. Buturlakin, Spectra of nite linear and unitary groups, Algebra Logic, 47(2) (2008), 91-99.
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Eynsham, 1985.
- O. H. King, The subgroup structure of nite classical groups in terms of geo- metric congurations, Surveys in combinatorics, London Math. Soc., Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, (2005), 29-56.
- A. V. Vasilev and E. P. Vdovin, An adjacency criterion in the prime graph of a nite simple group, Algebra Logic, 44(6) (2005), 381-406.
- A. V. Vasilev and E. P. Vdovin, Cocliques of maximal size in the prime graph of a nite simple group, Algebra Logic, 50(4) (2011), 291-322.
- A. V. Zavarnitsin and V. D. Mazurov, Element orders in coverings of symmet- ric and alternating groups, Algebra Logic, 38(3) (1999), 159-170.
- K. Zsigmondy, Zur theorie der potenzreste, Monatsh. Math. Phys., 3(1) (1892) 265-284.

Year 2018,
Volume: 24 Issue: 24, 73 - 90, 05.07.2018
### Abstract

### References

The solvable graph of a nite group G, which is denoted by

s(G), is a simple graph whose vertex set is comprised of the prime divisors

of jGj 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 p1 < p2 < < pk be all prime divisors of jGj and let

Ds(G) = (ds(p1); ds(p2); : : : ; ds(pk)), where ds(p) signies the degree of the

vertex p in s(G). We will simply call Ds(G) the degree pattern of solvable

graph of G. A nite group H is said to be ODs-characterizable if H = G for

every nite group G such that jGj = jHj and Ds(G) = Ds(H). In this paper,

we study the solvable graph of some subgroups and some extensions of a nite

group. Furthermore, we prove that the linear groups L3(q) with certain properties,

are ODs-characterizable

- S. Abe, A characterization of some nite simple groups by orders of their solvable subgroups, Hokkaido Math. J., 31(2) (2002), 349-361.
- S. Abe and N. Iiyori, A generalization of prime graphs of nite groups, Hokkaido Math. J., 29(2) (2000), 391-407.
- B. Akbari, N. Iiyori and A. R. Moghaddamfar, A new characterization of some simple groups by order and degree pattern of solvable graph, Hokkaido Math. J., 45(3) (2016), 337-363.
- J. N. Bray, D. F. Holt and C. M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Mathematical Society Lecture Note Series, 407, Cambridge University Press, Cambridge, 2013.
- A. A. Buturlakin, Spectra of nite linear and unitary groups, Algebra Logic, 47(2) (2008), 91-99.
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Eynsham, 1985.
- O. H. King, The subgroup structure of nite classical groups in terms of geo- metric congurations, Surveys in combinatorics, London Math. Soc., Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, (2005), 29-56.
- A. V. Vasilev and E. P. Vdovin, An adjacency criterion in the prime graph of a nite simple group, Algebra Logic, 44(6) (2005), 381-406.
- A. V. Vasilev and E. P. Vdovin, Cocliques of maximal size in the prime graph of a nite simple group, Algebra Logic, 50(4) (2011), 291-322.
- A. V. Zavarnitsin and V. D. Mazurov, Element orders in coverings of symmet- ric and alternating groups, Algebra Logic, 38(3) (1999), 159-170.
- K. Zsigmondy, Zur theorie der potenzreste, Monatsh. Math. Phys., 3(1) (1892) 265-284.

There are 11 citations in total.

Primary Language | English |
---|---|

Subjects | Mathematical Sciences |

Journal Section | Articles |

Authors | |

Publication Date | July 5, 2018 |

Published in Issue | Year 2018 Volume: 24 Issue: 24 |