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| Paper IPM / M / 11866 |
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| Abstract: | |
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The degree pattern of a finite group M has been
introduced in []. 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. It is shown that the alternating groups
Am and Am+1, for m=27, 35, 51, 57, 65, 77,
87, 93 and 95, are OD-characterizable, while their
automorphism groups are 3-fold OD-characterizable. It is also
shown that the symmetric groups Sm+2, for m=7, 13, 19,
23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83,
89 and 97, are 3-fold OD-characterizable. 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 Am are
OD-characterizable, while the symmetric groups Sm are
OD-characterizable or 3-fold OD-characterizable; (b) The
alternating group A10 is 2-fold OD-characterizable (c)
The symmetric group S10 is 8-fold OD-characterizable. This
theorem completes the study of OD-characterizability of the
alternating and symmetric groups Am and Sm of degree m ≤ 100.
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