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
The Lie algebra of derivations of rational function field C(t) is C(t) dt/d.The automorphism group of C(t) is well known as to be isomorphic to the projective linear groupPGL(2, C). In this short note we prove that ev...The Lie algebra of derivations of rational function field C(t) is C(t) dt/d.The automorphism group of C(t) is well known as to be isomorphic to the projective linear groupPGL(2, C). In this short note we prove that every automorphism of C(t) dt/d can be induced in anatural way from an automorphism of C(t).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.
文摘The Lie algebra of derivations of rational function field C(t) is C(t) dt/d.The automorphism group of C(t) is well known as to be isomorphic to the projective linear groupPGL(2, C). In this short note we prove that every automorphism of C(t) dt/d can be induced in anatural way from an automorphism of C(t).
