# Subhypergroup

## Definition

### Definition with symbols

A subset $H$ of a hypergroup $K$ is said to be a subhypergroup if $H$ satisfies the following three conditions:

• The identity element of $K$ lies inside $H$
• The involute of any element of $H$ lies inside $H$
• The convolution of any two probability measures that lie only inside $H$ also lies inside $H$

### Analogy with definition of subgroup

The three conditions for a subhypergroup correspond closely with the three conditions for a subgroup.

• The requirement that the identity element of the hypergroup lie inside the subset corresponds to the requirement that the identity element of the group lie inside the subset
• The requirement that the involute of any element of the subset also lie inside the subset corresponds to the requirement that the inverse of any element in the subset (of the group) also lie inside the subset
• The requirement that the convolution of probability measures inside the subset also lie inside the subset corresponds to the requirement that the product of any two elements inside the subset (of the group) also lie inside the subset