For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic de...For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.展开更多
Let p be a prime and F be a finite field of characteristic p.Suppose that FG is the group algebra of the finite p-group G over the field F.Let V(FG)denote the group of normalized units in FG and let V_(*)(FG)denote th...Let p be a prime and F be a finite field of characteristic p.Suppose that FG is the group algebra of the finite p-group G over the field F.Let V(FG)denote the group of normalized units in FG and let V_(*)(FG)denote the unitary subgroup of V(FG).If p is odd,then the order of V_(*)(FG)is|F|^((|G-1)/2).However,the case p=2 still is open.In this paper,the order of V*(FG)is computed when G is a nonabelian 2-group given by a central extension of the form 1→Z_(2^(n))×Z_(2^(m))→G→Z_(2)×…×Z_(2)→1 and G'≌Z_(2),n,m≥1.Furthermore,a conjecture is confirmed,i.e.,the order of V_(*)(FG)can be divisible by|F|^(1/2(|G|+|Ω1(G)|)-1),where Ω_(1)(G)={g∈G|g^(2)=1}.展开更多
Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant ...Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.展开更多
Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their ...Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator qu-otient, the information content of each coclass subtree with metabelian main-line is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.展开更多
In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {...In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {α∈ AutG | α acts trivially on G }. Then Aut G G⊿AutG and AutG/Aut G G≌Z p-1 . Furthermore, (i) If G is of exponent p m , then Aut G G/InnG≌Sp(2n, p) × Z p m-1 . (ii) If G is of exponent p m+1 , then Aut G G/InnG≌ (K Sp(2n-2, p))×Z p m-1 , where K is an extraspecial p-group of order p 2n-1 . In particular, Aut G G/InnG≌ Z p × Z p m-1 when n = 1. (2) When p = 2, then, (i) If G is of exponent 2 m , then AutG≌ Sp(2n, 2) × Z 2 × Z 2 m-2 . In particular, when n = 1, |AutG| = 3 · 2 m+2 . None of the Sylow subgroups of AutG is normal, and each of the Sylow 2-subgroups of AutG is isomorphic to H K, where H = Z 2 × Z 2 × Z 2 × Z 2 m-2 , K = Z 2 . (ii) If G is of exponent 2 m+1 , then AutG≌ (I Sp(2n-2, 2)) × Z 2 × Z 2 m-2 , where I is an elementary abelian 2-group of order 2 2n-1 . In particular, when n = 1, |AutG| = 2 m+2 and AutG≌ H K, where H = Z 2 × Z 2 × Z 2 m-1 , K = Z 2 .展开更多
In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2^3. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of P...In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2^3. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.展开更多
Assume G is a group of order 2^(n),n≥5.Let s_(k)(G)denote the number of subgroups of order 2^(k) of G.We classify finite 2-groups G with s k(G)≤2^(4),where 1≤k≤n.
从极大子群、中心主因子和正规子群的G 主列的角度来讨论有限幂零群,获得了有限幂零群的若干新刻划.设M是有限群G的任一极大子群,H G.令 G=G/Φ(G),则G是幂零群当且仅当下列条件之一成立:(1)如H≤\M,则H∩M G且H/H∩M≤Z(G/H∩M);(2)如H...从极大子群、中心主因子和正规子群的G 主列的角度来讨论有限幂零群,获得了有限幂零群的若干新刻划.设M是有限群G的任一极大子群,H G.令 G=G/Φ(G),则G是幂零群当且仅当下列条件之一成立:(1)如H≤\M,则H∩M G且H/H∩M≤Z(G/H∩M);(2)如H≤\M,则M≤CG(H/H∩M);(3)如H≤\M,则H≤CG(M/H∩M);(4)如H≤\M,则M补于G的一个中心主因子;(5)F( G)有一个 G 主列,其中每个主因子都是 G中心的且CG(F( G))可解;(6)Soc( G)有一个 G 主列,其中每个主因子都是 G 中心的;(7)K∞(G)≤H,H/Φ(H)有一个 G 主列其中每个主因子都是 G 中心的;(8)HCG(H)≤Z∞(G).展开更多
基金Supported by the NSF of China(11171194)by the NSF of Shanxi Province(2012011001-1)
文摘For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.
基金supported by the National Natural Science Foundation of China(Nos.11771129,11301150,11601121)the Natural Science Foundation of Henan Province of China(No.162300410066)
文摘Let G be a finite p-group with a cyclic Frattini subgroup. In this paper, the automorphism group of G is determined.
基金supported by National Natural Science Foundation of China(Grant No.12171142)。
文摘Let p be a prime and F be a finite field of characteristic p.Suppose that FG is the group algebra of the finite p-group G over the field F.Let V(FG)denote the group of normalized units in FG and let V_(*)(FG)denote the unitary subgroup of V(FG).If p is odd,then the order of V_(*)(FG)is|F|^((|G-1)/2).However,the case p=2 still is open.In this paper,the order of V*(FG)is computed when G is a nonabelian 2-group given by a central extension of the form 1→Z_(2^(n))×Z_(2^(m))→G→Z_(2)×…×Z_(2)→1 and G'≌Z_(2),n,m≥1.Furthermore,a conjecture is confirmed,i.e.,the order of V_(*)(FG)can be divisible by|F|^(1/2(|G|+|Ω1(G)|)-1),where Ω_(1)(G)={g∈G|g^(2)=1}.
文摘Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.
文摘Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator qu-otient, the information content of each coclass subtree with metabelian main-line is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.
基金supported by National Natural Science Foundation of China (Grant No.10671058)Doctor Foundation of Henan University of Technology (Grant No. 2009BS029)
文摘In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {α∈ AutG | α acts trivially on G }. Then Aut G G⊿AutG and AutG/Aut G G≌Z p-1 . Furthermore, (i) If G is of exponent p m , then Aut G G/InnG≌Sp(2n, p) × Z p m-1 . (ii) If G is of exponent p m+1 , then Aut G G/InnG≌ (K Sp(2n-2, p))×Z p m-1 , where K is an extraspecial p-group of order p 2n-1 . In particular, Aut G G/InnG≌ Z p × Z p m-1 when n = 1. (2) When p = 2, then, (i) If G is of exponent 2 m , then AutG≌ Sp(2n, 2) × Z 2 × Z 2 m-2 . In particular, when n = 1, |AutG| = 3 · 2 m+2 . None of the Sylow subgroups of AutG is normal, and each of the Sylow 2-subgroups of AutG is isomorphic to H K, where H = Z 2 × Z 2 × Z 2 × Z 2 m-2 , K = Z 2 . (ii) If G is of exponent 2 m+1 , then AutG≌ (I Sp(2n-2, 2)) × Z 2 × Z 2 m-2 , where I is an elementary abelian 2-group of order 2 2n-1 . In particular, when n = 1, |AutG| = 2 m+2 and AutG≌ H K, where H = Z 2 × Z 2 × Z 2 m-1 , K = Z 2 .
文摘In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2^3. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.
文摘Assume G is a group of order 2^(n),n≥5.Let s_(k)(G)denote the number of subgroups of order 2^(k) of G.We classify finite 2-groups G with s k(G)≤2^(4),where 1≤k≤n.
文摘从极大子群、中心主因子和正规子群的G 主列的角度来讨论有限幂零群,获得了有限幂零群的若干新刻划.设M是有限群G的任一极大子群,H G.令 G=G/Φ(G),则G是幂零群当且仅当下列条件之一成立:(1)如H≤\M,则H∩M G且H/H∩M≤Z(G/H∩M);(2)如H≤\M,则M≤CG(H/H∩M);(3)如H≤\M,则H≤CG(M/H∩M);(4)如H≤\M,则M补于G的一个中心主因子;(5)F( G)有一个 G 主列,其中每个主因子都是 G中心的且CG(F( G))可解;(6)Soc( G)有一个 G 主列,其中每个主因子都是 G 中心的;(7)K∞(G)≤H,H/Φ(H)有一个 G 主列其中每个主因子都是 G 中心的;(8)HCG(H)≤Z∞(G).
