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Paper IPM / M / 11866  


Abstract:  
The degree pattern of a finite group M has been
introduced in []. A group M is called kfold
ODcharacterizable if there exist exactly k nonisomorphic
finite groups having the same order and degree pattern as M. In
particular, a 1fold ODcharacterizable group is simply called
ODcharacterizable. It is shown that the alternating groups
A_{m} and A_{m+1}, for m=27, 35, 51, 57, 65, 77,
87, 93 and 95, are ODcharacterizable, while their
automorphism groups are 3fold ODcharacterizable. It is also
shown that the symmetric groups S_{m+2}, for m=7, 13, 19,
23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83,
89 and 97, are 3fold ODcharacterizable. From this, the
following theorem is derived. Let m be a natural number such
that m ≤ 100. Then one of the following holds: (a) if
m ≠ 10, then the alternating groups A_{m} are
ODcharacterizable, while the symmetric groups S_{m} are
ODcharacterizable or 3fold ODcharacterizable; (b) The
alternating group A_{10} is 2fold ODcharacterizable (c)
The symmetric group S_{10} is 8fold ODcharacterizable. This
theorem completes the study of ODcharacterizability of the
alternating and symmetric groups A_{m} and S_{m} of degree m ≤ 100.
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