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Finite groups which have many normal subgroups
Qinhai Zhang,Xiaoqiang Guo,Haipeng Qu,Mingyao Xu 대한수학회 2009 대한수학회지 Vol.46 No.6
In this paper we classify finite groups whose nonnormal subgroups are of order p or pq, where p, q are primes. As a by-product, we also classify the finite groups in which all nonnormal subgroups are cyclic. In this paper we classify finite groups whose nonnormal subgroups are of order p or pq, where p, q are primes. As a by-product, we also classify the finite groups in which all nonnormal subgroups are cyclic.
NOMALIZERS OF NONNORMAL SUBGROUPS OF FINITE p-GROUPS
Zhang, Qinhai,Gao, Juan Korean Mathematical Society 2012 대한수학회지 Vol.49 No.1
Assume G is a finite p-group and i is a fixed positive integer. In this paper, finite p-groups G with ${\mid}N_G(H):H{\mid}=p^i$ for all nonnormal subgroups H are classified up to isomorphism. As a corollary, this answer Problem 116(i) proposed by Y. Berkovich in his book "Groups of Prime Power Order Vol. I" in 2008.
FINITE GROUPS WHICH HAVE MANY NORMAL SUBGROUPS
Zhang, Qinhai,Guo, Xiaoqiang,Qu, Haipeng,Xu, Mingyao Korean Mathematical Society 2009 대한수학회지 Vol.46 No.6
In this paper we classify finite groups whose nonnormal subgroups are of order p or pq, where p, q are primes. As a by-product, we also classify the finite groups in which all nonnormal subgroups are cyclic.
NOMALIZERS OF NONNORMAL SUBGROUPS OF FINITE p-GROUPS
Qinhai Zhang,Juan Gao 대한수학회 2012 대한수학회지 Vol.49 No.1
Assume G is a finite p-group and i is a fixed positive integer. In this paper, finite p-groups G with jNG(H) : Hj = pi for all nonnormal subgroups H are classified up to isomorphism. As a corollary, this answers Problem 116(i) proposed by Y. Berkovich in his book "Groups of Prime Power Order Vol. I" in 2008. Assume G is a finite p-group and i is a fixed positive integer. In this paper, finite p-groups G with jNG(H) : Hj = pi for all nonnormal subgroups H are classified up to isomorphism. As a corollary, this answers Problem 116(i) proposed by Y. Berkovich in his book "Groups of Prime Power Order Vol. I" in 2008.
FINITE p-GROUPS ALL OF WHOSE SUBGROUPS OF CLASS 2 ARE GENERATED BY TWO ELEMENTS
Li, Pujin,Zhang, Qinhai Korean Mathematical Society 2019 대한수학회지 Vol.56 No.3
We proved that finite p-groups in the title coincide with finite p-groups all of whose non-abelian subgroups are generated by two elements. Based on the result, finite p-groups all of whose subgroups of class 2 are minimal non-abelian (of the same order) are classified, respectively. Thus two questions posed by Berkovich are solved.
Finite $p$-groups all of whose subgroups of class 2 are generated by two elements
Pujin Li,Qinhai Zhang 대한수학회 2019 대한수학회지 Vol.56 No.3
We proved that finite $p$-groups in the title coincide with finite $p$-groups all of whose non-abelian subgroups are generated by two elements. Based on the result, finite $p$-groups all of whose subgroups of class 2 are minimal non-abelian (of the same order) are classified, respectively. Thus two questions posed by Berkovich are solved.