The Answers to a Problem and Two Conjectures about OD-Characterization of Finite Groups

In [Akbari and Moghaddamfar, Recognizing by order and degree pattern of some projective special linear groups, {\it Internat. J. Algebra Comput.}, 2012] the authors possed the following problem: \\ {\bf Problem.} {\it Is there a simple group which is $k$-fold OD-characterizable for $k\geq3\ ?$ } In this paper as the main result we give positive answer to the above problem and we introduce two simple groups which are $k$-fold OD-characterizable such that $k\geq6$. Also in [R. Kogani-Moghadam and A. R. Moghaddamfar, Groups with the same order and degree pattern, {\it Science China Mathematics}, 2012], the authors possed two conjectures as follows: \\ {\bf Conjecture 1.} {\it All alternating groups $A_m$ with $m \not= 10$ are OD-characterizable.} \\ {\bf Conjecture 2.} {\it All symmetric groups $S_m$, with $m \not= 10$, are $n$-fold OD-characterizable, where $n\in\{1, 3\}$.} In this paper we find some alternating and some symmetric groups such that these conjectures are not true for them.