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 ...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＋l, 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 Sin＋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 ODcharacterizable 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.展开更多
基金partially supported by a research grant fromthe Institute for Research in Fundamental Sciences (IPM)
文摘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＋l, 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 Sin＋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 ODcharacterizable 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.
