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The degree pattern of a finite group $M$ has been
introduced by A. R. Moghaddamfar et al. [Algebra Colloquium,
2005, 12(3): 431?442].
A group $M$ is called $k$-fold OD-characterizable if
there exist exactly $k$ non-isomorphic finite groups having the
same order and degree
pattern as $M$. In particular, a 1-fold OD-characterizable group
is simply called OD-characterizable. In this article, we
will show that the alternating groups $A_{p+3}$ for $p=23, 31,
37, 43$ and $47$ are OD-characterizable. Moreover, we show that
the automorphism groups of these groups are 3-fold
OD-characterizable. It is worth mentioning that the prime graphs
associated with all these groups are connected.
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