Product of subgroups: Difference between revisions

Latest revision as of 13:24, 8 November 2023

Definition

Symbol-free definition

The Product of two subgroups of a group is the subset consiting of the pairwise products between the two subgroups.

Definition with symbols

The Product $HK$ of two subgroups $H$ and $K$ of a group $G$ is:

$HK:=\{hk\mid h\in H,k\in K\}$ .

If $G$ is abelian and if the group operation is denoted as $+$ , the product is termed the sum, and is denoted $H+K$ :

$H+K=\{h+k\mid h\in H,k\in K\}$ .

Facts

$HK$ is the double coset $HeK$ , $e$ being the identity element of $G$ .

The product $HK$ is in general not a subgroup, because it may not be closed under the group operation.

The smallest subgroup containing $HK$ is the join $\langle H,K\rangle$ of $H$ and $K$ , which is also the subgroup generated by $H$ and $K$ .

Following statements are equivalent:

$HK$ is a subgroup
$HK=\langle H,K\rangle$ , viz., it is precisely the join of $H$ and $K$ (the subgroup generated by $H$ and $K$ )
$\!HK=KH$
$HK\subseteq KH$
$KH\subseteq HK$

If the above equivalent conditions hold, $H$ and $K$ are termed permuting subgroups.

@@ Line 1: / Line 1: @@
 ==Definition==
-Let <math>G</math> be a [[group]] and <math>H,K</math> be [[subgroup]]s of <math>G</math>. Then the '''product''' of <math>H</math> and <math>K</math> is defined as the set of elements:
+===Symbol-free definition===
-<math>\{ hk | h \in H, k \in K \}</math>
+The '''Product''' of two [[subgroup]]s of a group is the subset consiting of the pairwise products between the two subgroups.
-The following are equivalent (but may not all in general be true):
+===Definition with symbols===
+The '''Product''' <math>HK</math> of two [[subgroup]]s <math>H</math> and <math>K</math> of a group <math>G</math> is:
+<math>HK := \{ hk \mid h \in H, k \in K \}</math>.
+If <math>G</math> is abelian and if the group operation is denoted as <math>+</math>, the ''product'' is termed the ''sum'', and is denoted <math>H + K</math>:
+<math>H + K = \{ h + k \mid h \in H, k \in K \}</math>.
+==Facts==
+<math>HK</math> is the [[double coset]] <math>HeK</math>, <math>e</math> being the identity element of <math>G</math>.
+The cardinality <math>\left| HK \right|</math> of <math>HK</math> is <math>\left| H \right| \left| K \right| / \left| H \cap K  \right|</math>.
+The product <math>HK</math> is in general not a subgroup, because it may not be closed under the group operation.
+The smallest subgroup containing <math>HK</math> is the [[join of subgroups|join]] <math>\langle H, K \rangle</math> of <math>H</math> and <math>K</math>, which is also the subgroup generated by <math>H</math> and <math>K</math>.
+Following statements are equivalent:
 * <math>HK</math> is a [[subgroup]]
-* <math>HK =  \langle H,K \rangle</math>, viz it is precisely the join of <math>H</math> and <math>K</math> (the subgroup generated by <math>H</math> and <math>K</math>)
+* <math>HK =  \langle H,K \rangle</math>, viz., it is precisely the join of <math>H</math> and <math>K</math> (the subgroup generated by <math>H</math> and <math>K</math>)
-* <math>HK = KH</math>
+* <math>\! HK = KH</math>
 * <math>HK \subseteq KH</math>
 * <math>KH \subseteq HK</math>
 If the above equivalent conditions hold, <math>H</math> and <math>K</math> are termed [[permuting subgroups]].