Throughout this paper, all groups are finite and G always denotes a finite group. We use π（G） to denote the set of all primes dividing ｜G｜. A subgroup A of G is said to permute with a subgroup B if AB = BA. In th...Throughout this paper, all groups are finite and G always denotes a finite group. We use π（G） to denote the set of all primes dividing ｜G｜. A subgroup A of G is said to permute with a subgroup B if AB = BA. In this case, we say also that the subgroups A and B are permutable.展开更多
A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur- Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset repr...A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur- Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall （STACS 2009）, we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential （in the number of coordinates） time algorithm for code isomorphism problem developed recently by Babai et al. （SODA 2011）. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups： given two representations ρandτ- of a group H over Zp^d, a prime, determine if there exists an automorphism ： ФH→ H, such that the induced representation p Ф= ρ o Ф and τ are equivalent, in time poly（｜H｜, p^d）.展开更多
In this paper, we give some new conditions of the existence of Hall subgroups in non-soluble finite groups, and so the famous Hall theorem and Schur-Zassenhaus theorem are generalized.
In the paper, we introduce some concepts and notations of Hall π-subgroup etc, and prove some properties about finite p-group, nilpotent group and Sylow p-subgroup. Finally, we have proved two interesting theorems ab...In the paper, we introduce some concepts and notations of Hall π-subgroup etc, and prove some properties about finite p-group, nilpotent group and Sylow p-subgroup. Finally, we have proved two interesting theorems about nilpotent subgroup.展开更多
文摘Throughout this paper, all groups are finite and G always denotes a finite group. We use π（G） to denote the set of all primes dividing ｜G｜. A subgroup A of G is said to permute with a subgroup B if AB = BA. In this case, we say also that the subgroups A and B are permutable.
基金supported in part by the National Natural Science Foundation of China under Grant No. 60553001the National Basic Research 973 Program of China under Grant Nos. 2007CB807900 and 2007CB807901
文摘A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur- Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall （STACS 2009）, we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential （in the number of coordinates） time algorithm for code isomorphism problem developed recently by Babai et al. （SODA 2011）. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups： given two representations ρandτ- of a group H over Zp^d, a prime, determine if there exists an automorphism ： ФH→ H, such that the induced representation p Ф= ρ o Ф and τ are equivalent, in time poly（｜H｜, p^d）.
基金The research is supported by the NNSF of China （Grant 11371335） and the Natural Science Foundation of Shandong Province （Grant ZR2014AL001）, China.
文摘In this paper, we give some new conditions of the existence of Hall subgroups in non-soluble finite groups, and so the famous Hall theorem and Schur-Zassenhaus theorem are generalized.
文摘In the paper, we introduce some concepts and notations of Hall π-subgroup etc, and prove some properties about finite p-group, nilpotent group and Sylow p-subgroup. Finally, we have proved two interesting theorems about nilpotent subgroup.
