Recognizing Finite Groups Through Order and Degree Patterns

The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree patterns: All sporadic simple groups, the alternating groups A _p (p ≥ 5 is a twin prime) and some simple groups of the Lie type. In this paper, the authors continue this investigation. In particular, the authors show that the symmetric groups S _p+3, where p + 2 is a composite number and p + 4 is a prime and 97 < p ∈ π(1000!), are 3-fold OD-characterizable. The authors also show that the alternating groups A116 and A134 are OD-characterizable. It is worth mentioning that the latter not only generalizes the results by Hoseini in 2010 but also gives a positive answer to a conjecture by Moghaddamfar in 2009.