1.
Let \(G = D_4\text{,}\) the symmetries of the square. Let \(H = \{r_0, r_2, f_1, f_2\}\text{.}\) Is \(H\) a subgroup of \(G\text{?}\)
Worksheet Introduction to Subgroups
A subgroup \(H\) of a group \(G\) is a group whose elements are all elements of \(G\) and whose operation is the operation of \(G\text{,}\) restricted to the elements of \(H\text{.}\) Thus given any subset of a group, we can ask if that subset is a subgroup.
2.
3.
Find two subsets of \(\Z_6\text{:}\) one a subgroup and one not a subgroup.
4.
Let \(G\) be any group and consider the set \(H = \{g \in G \st g = g\inv\}\text{.}\) Must \(H\) be a subgroup of \(G\text{?}\) Explain why or find a counterexample.
Let \(G\) be a group and \(H\) a subgroup of \(G\text{.}\) Consider the following subsets of \(G\text{:}\)
\begin{equation*}
Z(G) = \{c \in G \st cx = xc \text{ for every } x \in G\}
\end{equation*}
\begin{equation*}
C(H) = \{g \in G \st ghg\inv = h \text{ for all } h \in H\}
\end{equation*}
\begin{equation*}
G^2 = \{g^2 \st g \in G\}\text{.}
\end{equation*}
5.
Consider the specific example \(G = D_4\) and \(H = \{r_0, r_2\}\text{.}\) Find \(Z(G)\text{,}\) \(C(H)\text{,}\) and \(G^2\text{.}\)
6.
Which of the sets you described above are subgroups of \(D_4\text{?}\)
7.
Prove that if \(G\) is any abelian group, then \(G^2\) is a subgroup of \(G\text{.}\) Where do you use the fact that \(G\) is abelian?
8.
Will \(Z(G)\) and \(C(H)\) always be subgroups of \(G\) for any group \(G\) and subgroup \(H\text{?}\) How do you know?
