Difference between revisions of "Centre of a group"
From Encyclopedia of Mathematics
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| + | The set $Z$ of all central elements (sometimes also called invariant elements) of the group, that is, the elements that commute with all elements of the group. The centre of a group $G$ is a [[normal subgroup]], and even a [[characteristic subgroup]] in $G$. Moreover, every subgroup of the centre is normal in $G$. Abelian groups and only these coincide with their centres. Groups whose centres consist only of the unit element are said to be groups without centre or groups with trivial centre. The quotient group $G/Z$ of a group $G$ by its centre is not necessarily a group without centre. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian)</TD></TR></table> | ||
Latest revision as of 18:26, 5 December 2014
The set $Z$ of all central elements (sometimes also called invariant elements) of the group, that is, the elements that commute with all elements of the group. The centre of a group $G$ is a normal subgroup, and even a characteristic subgroup in $G$. Moreover, every subgroup of the centre is normal in $G$. Abelian groups and only these coincide with their centres. Groups whose centres consist only of the unit element are said to be groups without centre or groups with trivial centre. The quotient group $G/Z$ of a group $G$ by its centre is not necessarily a group without centre.
References
| [1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |
How to Cite This Entry:
Centre of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centre_of_a_group&oldid=15435
Centre of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centre_of_a_group&oldid=15435
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article