• • • • • • • ## “School of Mathematics”

Back to Papers Home
Back to Papers of School of Mathematics

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
Abstract:
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.        