2E: Exercises

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\(H(C) =\Bigg\{ \begin{bmatrix}1 & a & b\\
0 & 1 & c \\
0 & 0 & 1 \end{bmatrix}\Big| \ a,b,c, \in \mathbb{C}\Bigg\}\).

Show that \(H(\mathbb{C})\) is a group under matrix multiplication. Demonstrate explicitly that \(H(\mathbb{C})\) is always non-abelian.

Let \(S = \mathbb{R} \backslash \{-1\}\) and define a binary operation \(\oplus\) on \(S\) by \(a \oplus b = a+b+ab\). Prove that \((S, \oplus)\) is an abelian group.
Given the groups \(\mathbb{R}^*\) and \(\mathbb{Z}\), let \(G = \mathbb{R}^* \times \mathbb{Z}\). Define a binary operation \(\star\) by \((a,m) \star (b, n) = (ab,m + n)\). Show that \( (G, \star)\) is a group under this operation.
Let \(G\) be a group. Show that if \(a^2 = e\), for all elements of \(a \in G\), then \(G\) must be abelian.

Let \(H\) consists of \(2 \times 2\) matrices of the form

\(\begin{bmatrix} \cos{(x)} & -\sin{(x)}\\
\sin{(x)} & \cos{(x)} \end{bmatrix}\), where \(x \in \mathbb{R}\). Prove that \(H\) is a subgroup of \(SL_2(\mathbb{R})\).

Prove or disprove the following statements: Let \(H\) and \(K\) be subgroups of a group \(G\).
1. \(H \cup K\) is a subgroup of \(G\).
2. \(H \cap K\) is a subgroup of \(G\).
Prove that for each element \(a \in G\), where \(G\) is a group, the centralizer of \(a\), \(C(a)\) is a subgroup of \(G\). Prove that for each element \(a \in G\), where \(G\) is a group, that \(C(a)=C(a^{-1})\).
Let \(G=GL(2,\mathbb{R})\). Then find
1. \(C \Bigg(\begin{bmatrix}1 & 1\\
  1 & 0 \end{bmatrix}\Bigg)\).
2. \(C \Bigg( \begin{bmatrix} 0 & 1\\
  1 & 0 \end{bmatrix}\Bigg)\).
3. \(\mathbb{Z}(G)\).

Let \(G\) be a group. \(H=\{g^2: g\in G\}. \)

Prove or disprove: If \(G\) is abelian then \(H\) is a subgroup.
Prove or disprove: If \(H\) is a subgroup then \(G\) is abelian.

List the cyclic subgroups of \(U(30) \).

\(U(30)=\{1,7,11,13,17,19,23,29\} \).

Thus \(|U(30)|=8 \).

Show that \(U(20)\ne <k> \) for any \(k \) in \(U(20) \). [Hence, \(U(20) \) is not cyclic.]

\(U(20)=\{1,3,7,9,11,13,17,19\} \).

\(|U(20)|=8 \).

Decide whether \(U(10) \) is cyclic or not.
Is \((\mathbb{Z},+) \) cyclic group? If so, what are the possible generators?