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Embed Dark Mode Prev Up Next \(\require{mathtools}\require{color} \setcounter{MaxMatrixCols}{15}
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Section Quotient Groups
Worksheet Part 1: Normal Subgroup Practice
The goal of these problems is to familiarize you with the definition of normal subgroup, the Normal Subgroup Test, and find more examples of normal subgroups.
1. Show that the subgroup
\(\SL(2,\R)\) of
\(\GL(2,\R)\) is normal in
\(\GL(2,\R)\text{.}\)
2. Show that the center of a group
\(G\) is normal in the group.
3. Suppose that
\(G\) has a unique subgroup
\(H\) of order
\(n\text{.}\) Show that
\(H\) is normal in
\(G\text{.}\)
4. Suppose that the index of a subgroup
\(H\) in a group
\(G\) is
\(2\text{.}\) Show that
\(H\) is normal in
\(G\text{.}\)
Hint .
Consider separately the cases
\(g\in H\) and
\(g \notin H\text{.}\) In the second case, you should consider proof by contradiction.
Worksheet Part 2: Quotient Practice
The goal of this problem is to practice working with cosets, quotients, and learn a theorem about quotients.
1.
Let
\(G = \Z_4 \oplus U(4)\) and consider the subgroups
\(H=\langle (2,3)\rangle\) and
\(K=\langle (2,1)\rangle\text{.}\)
(a) Show that
\(H\cong K\text{.}\)
(b) Compute the quotient
\(G/H\text{.}\) (Why is
\(H\) normal in
\(G\text{?}\) )
(c) Compute the quotient
\(G/K\text{.}\) (Why is
\(K\) normal in
\(G\text{?}\) )
(d) Show that
\(G/H \not\cong G/K\text{.}\)
(e)
True/False: If
\(G\) is a group with normal subgroups
\(H\) and
\(K\) and
\(H\cong K\text{,}\) then
\(G/H \cong G/K\text{.}\)
