On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups
Let G be a finite group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |𝒩𝒞(G)|≤2, then G is solvable, and (2) G is a nonsolvable group with |𝒩𝒞(G)|=3 if and only if G≅PSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13).
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2013-01-01
|
| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2013/430870 |
| id |
doaj-730e850cac314bba85bc34f9830ad651 |
|---|---|
| record_format |
Article |
| spelling |
doaj-730e850cac314bba85bc34f9830ad6512020-11-25T01:53:46ZengHindawi LimitedThe Scientific World Journal1537-744X2013-01-01201310.1155/2013/430870430870On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite GroupsJiangtao Shi0Cui Zhang1School of Mathematics and Information Science, Yantai University, Yantai 264005, ChinaDepartment of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Lyngby, DenmarkLet G be a finite group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |𝒩𝒞(G)|≤2, then G is solvable, and (2) G is a nonsolvable group with |𝒩𝒞(G)|=3 if and only if G≅PSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13).http://dx.doi.org/10.1155/2013/430870 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Jiangtao Shi Cui Zhang |
| spellingShingle |
Jiangtao Shi Cui Zhang On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups The Scientific World Journal |
| author_facet |
Jiangtao Shi Cui Zhang |
| author_sort |
Jiangtao Shi |
| title |
On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups |
| title_short |
On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups |
| title_full |
On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups |
| title_fullStr |
On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups |
| title_full_unstemmed |
On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups |
| title_sort |
on the set of the numbers of conjugates of noncyclic proper subgroups of finite groups |
| publisher |
Hindawi Limited |
| series |
The Scientific World Journal |
| issn |
1537-744X |
| publishDate |
2013-01-01 |
| description |
Let G be a finite group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |𝒩𝒞(G)|≤2, then G is solvable, and (2) G is a nonsolvable group with |𝒩𝒞(G)|=3 if and only if G≅PSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13). |
| url |
http://dx.doi.org/10.1155/2013/430870 |
| work_keys_str_mv |
AT jiangtaoshi onthesetofthenumbersofconjugatesofnoncyclicpropersubgroupsoffinitegroups AT cuizhang onthesetofthenumbersofconjugatesofnoncyclicpropersubgroupsoffinitegroups |
| _version_ |
1724989192379826176 |