Groups with the same order and degree pattern

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2012,（4）
• 页码范围: 701-720
• 页数: 20页
• 语种: 英文
• DOI: 10.1007/sl1425-011-4314-6

摘要

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.