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Paper   IPM / M / 11866
School of Mathematics
  Title:   Groups with the same order and degree pattern
  Author(s):  A. R. Moghaddamfar (Joint with R. Kogani-Moghaddam)
  Status:   Published
  Journal: Science China Mathematics
  Vol.:  55
  Year:  2012
  Pages:   701-720
  Supported by:  IPM
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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