Difference between revisions of "Carter subgroup"
From Encyclopedia of Mathematics
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| − | A maximal nilpotent subgroup of a group that coincides with its normalizer. Introduced by R. Carter [[#References|[1]]]. Any finite solvable group | + | {{TEX|done}} |
| + | A maximal nilpotent subgroup of a group that coincides with its normalizer. Introduced by R. Carter [[#References|[1]]]. Any finite solvable group $G$ has a Carter subgroup, and all Carter subgroups of $G$ are conjugate (Carter's theorem). | ||
====References==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
| − | An example of a non-solvable group having no Carter subgroup is | + | An example of a non-solvable group having no Carter subgroup is $A_5$, the alternating group of order 5. |
Any Carter subgroup of a finite solvable group is a maximal nilpotent subgroup. | Any Carter subgroup of a finite solvable group is a maximal nilpotent subgroup. | ||
Latest revision as of 19:08, 17 August 2014
A maximal nilpotent subgroup of a group that coincides with its normalizer. Introduced by R. Carter [1]. Any finite solvable group $G$ has a Carter subgroup, and all Carter subgroups of $G$ are conjugate (Carter's theorem).
References
| [1] | R.W. Carter, "Nilpotent selfnormalizing subgroups of soluble groups" Math. Z. , 75 : 2 (1961) pp. 136–139 |
| [2] | A.I. Kostrikin, "Finite groups" Itogi Nauk. Algebra 1964 (1966) pp. 7–46 (In Russian) |
Comments
An example of a non-solvable group having no Carter subgroup is $A_5$, the alternating group of order 5.
Any Carter subgroup of a finite solvable group is a maximal nilpotent subgroup.
References
| [a1] | B. Huppert, "Endliche Gruppen" , 1 , Springer (1979) pp. 482–490 |
How to Cite This Entry:
Carter subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carter_subgroup&oldid=32982
Carter subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carter_subgroup&oldid=32982
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article