RECOGNIZING BY ORDER AND DEGREE PATTERN OF SOME PROJECTIVE SPECIAL LINEAR GROUPS

Let M be a finite group and D(M) be the degree pattern of M. Denote by h(OD)(M) the number of isomorphism classes of finite groups G with the same order and degree pattern as M. A finite group M is called k-fold OD-characterizable if h(OD)(M) = k. Usually, a 1-fold OD-characterizable group is simply called OD-characterizable. The purpose of this article is twofold. First, it provides some information on the structure of a group from its degree pattern. Second, it proves that the projective special linear groups L-4(4), L-4(8), L-4(9), L-4(11), L-4(13), L-4(16), L-4(17) are OD-characterizable.