---
tag: 0NS1YTGM19
kind: definition
name: Product of two subgroups
aliases: [frobenius-product]
msc: [20A05]
defines: 0PT52XQZD7
uses: [0N0PXCTYA7]
sources:
  - url: https://en.wikipedia.org/wiki/Product_of_group_subsets
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Product_of_group_subsets
    locator: Product of subgroups
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ and $K$ be [[subgroups|0X1JXT47GA]] 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|0RP7VWQ0YN]] $H \cup K$.

## Explanations

### L1 human

To multiply two subgroups, multiply each element of the first by each element of the second.
The result contains both subgroups, but it may fail to be a subgroup, because a product $hk h'k'$ need not have the form $h''k''$.
In an abelian group, the product of two subgroups is always a subgroup.