## Section3.3Subgroups

### Subsection3.3.1Definitions and Examples

Sometimes we wish to investigate smaller groups sitting inside a larger group. The set of even integers $2{\mathbb Z} = \{\ldots, -2, 0, 2, 4, \ldots \}$ is a group under the operation of addition. This smaller group sits naturally inside of the group of integers under addition. We define a subgroup $H$ of a group $G$ to be a subset $H$ of $G$ such that when the group operation of $G$ is restricted to $H\text{,}$ $H$ is a group in its own right. Observe that every group $G$ with at least two elements will always have at least two subgroups, the subgroup consisting of the identity element alone and the entire group itself. The subgroup $H = \{ e \}$ of a group $G$ is called the trivial subgroup. A subgroup that is a proper subset of $G$ is called a proper subgroup. In many of the examples that we have investigated up to this point, there exist other subgroups besides the trivial and improper subgroups.

### Subsection3.3.2Some Subgroup Theorems

Let us examine some criteria for determining exactly when a subset of a group is a subgroup.

##### Remark3.3.9Sage

The first half of this text is about group theory. Sage includes Groups, Algorithms and Programming (GAP), a program designed primarly for just group theory, and in continuous development since 1986. Many of Sage's computations for groups ultimately are performed by GAP.