Permuting subgroups: Difference between revisions

Latest revision as of 23:06, 15 March 2009

This article defines a symmetric relation on the collection of subgroups inside the same group.

Two subgroups $H$ and $K$ of a group $G$ are termed permuting subgroups if the following equivalent conditions hold:

$H K = K H$
$H K$ (the product of subgroups) is a subgroup
Given elements $h$ in $H$ and $k$ in $K$ , there exist elements $k^{'}$ in $K$ and $h^{'}$ in $H$ such that $h k = k^{'} h^{'}$ . In other words, $H K \subseteq K H$ .
$[H, K] \subseteq H K$ . In other words, the commutator of $H$ and $K$ is contained in their product.

@@ Line 1: / Line 1: @@
 {{subgroup symmrel}}
+<section begin="beginner"/>
 ==Definition==
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 Two subgroups <math>H</math> and <math>K</math> of a group <math>G</math> are termed '''permuting subgroups''' if the following equivalent conditions hold:
-* <math>HK = KH</math>
+# <math>HK = KH</math>
-* <math>HK</math> (the [[product of subgroups|product]]) is a subgroup
+# <math>HK</math> (the [[defining ingredient::product of subgroups]]) is a subgroup
-* Given elements <math>h</math> in <math>H</math> and <math>k</math> in <math>K</math>, there exist elements <math>k'</math> in <math>K</math> and <math>h'</math> in <math>H</math> such that <math>hk = k'h'</math>.
+# Given elements <math>h</math> in <math>H</math> and <math>k</math> in <math>K</math>, there exist elements <math>k'</math> in <math>K</math> and <math>h'</math> in <math>H</math> such that <math>hk = k'h'</math>. In other words, <math>HK \subseteq KH</math>.
+# <math>[H,K] \subseteq HK</math>. In other words, the [[defining ingredient::commutator of two subgroups|commutator]] of <math>H</math> and <math>K</math> is contained in their product.
 ===Equivalence of definitions===
 {{proofat|[[Equivalence of definitions of permuting subgroups]]}}
+<section end="beginner"/>
 ==Relation with other relations==