The degree pattern of a finite group G associated to its prime graph has been introduced in [IJ and it is proved that the following simple groups are uniquely determined by their degree patterns and orders: all sporadic simple groups , alternating groups .4.p (p 2: 5 is a twin prime) and some simple groups of Lie type. In the present paper. we continue this investigation. In particular, we show that the alternating groups .4.106 and A1l2 are OD-characterizable. We will also show that the symmetric groups 5 106 and 5112 are 3-fold OD-characterizable.
The main aim of this article is to characterize the finite simple groups by less character quantity. In fact, we show that each Mathieu-group G can be determined by their largest and second largest irreducible character degrees. MSC: 20C15
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