Group theory and generalizations Definition

Product of two subgroups

Definition

Let $G$ be a group and let $H$ and $K$ be subgroups of $G$ . The product of $H$ and $K$ is the subset $HK = \{hk : h \in H, k \in K\}$ of $G$ .

Remarks

The set $HK$ is a subgroup of $G$ if and only if $HK = KH$ . It need not be a subgroup. In the symmetric group $S_3$ , two distinct subgroups of order $2$ have a product with $4$ elements, and $4$ does not divide $6$ . When $HK$ is a subgroup, it is the subgroup generated by $H \cup K$ .

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Group concept
Subgroup definition
Product of two subgroups definition

Where this leads

13 pages build on this one.

Tag	`0NS1YTGM19`
URI	`m2://0NS1YTGM19`
Aliases	`frobenius-product`
Kind	definition
Writing	W0 stub
MSC	20A05
Defines	Product of two subgroups
Uses	Subgroup

Product of two subgroups

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by