Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements

Changguo Shao; Qinhui Jiang

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 3, page 827-831
  • ISSN: 0011-4642

Abstract

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Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups G whose set of numbers of subgroups of possible orders n ( G ) has exactly two elements. We show that if G is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then G has a normal Sylow subgroup of prime order and G is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with n ( G ) = { 1 , q + 1 } for some prime q . In particular, G is supersolvable under this condition.

How to cite

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Shao, Changguo, and Jiang, Qinhui. "Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements." Czechoslovak Mathematical Journal 64.3 (2014): 827-831. <http://eudml.org/doc/262152>.

@article{Shao2014,
abstract = {Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups $G$ whose set of numbers of subgroups of possible orders $n(G)$ has exactly two elements. We show that if $G$ is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then $G$ has a normal Sylow subgroup of prime order and $G $ is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with $n(G)=\lbrace 1, q+1\rbrace $ for some prime $q$. In particular, $G$ is supersolvable under this condition.},
author = {Shao, Changguo, Jiang, Qinhui},
journal = {Czechoslovak Mathematical Journal},
keywords = {finite group; number of subgroups of possible orders; finite groups; numbers of subgroups; possible orders of subgroups; conjugacy class sizes},
language = {eng},
number = {3},
pages = {827-831},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements},
url = {http://eudml.org/doc/262152},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Shao, Changguo
AU - Jiang, Qinhui
TI - Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 827
EP - 831
AB - Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups $G$ whose set of numbers of subgroups of possible orders $n(G)$ has exactly two elements. We show that if $G$ is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then $G$ has a normal Sylow subgroup of prime order and $G $ is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with $n(G)=\lbrace 1, q+1\rbrace $ for some prime $q$. In particular, $G$ is supersolvable under this condition.
LA - eng
KW - finite group; number of subgroups of possible orders; finite groups; numbers of subgroups; possible orders of subgroups; conjugacy class sizes
UR - http://eudml.org/doc/262152
ER -

References

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