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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.

\begin{equation*} \begin{array}{c|cccc} + & (0,0) & (0,1) & (1,0) & (1,1) \\ \hline (0,0) & (0,0) & (0,1) & (1,0) & (1,1) \\ (0,1) & (0,1) & (0,0) & (1,1) & (1,0) \\ (1,0) & (1,0) & (1,1) & (0,0) & (0,1) \\ (1,1) & (1,1) & (1,0) & (0,1) & (0,0) \end{array} \end{equation*}
Figure3.3.6Addition table for \({\mathbb Z}_2 \times {\mathbb Z}_2\)

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.