Groups with the same order and degree pattern

Abstract

The degree pattern of a finite group has been introduced in [18]. 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.