MCQ 1
Let \(G\) be a group and \(a,b\in G\) such that \(a^2=b^3=e_G.\) Then the inverse of \(ab^2ab\) is given by \(b^2aba.\)
- True
- False
MCQ 2
The set \(G=\{\mathbf{A}\in M_{n,n}(\mathbb{R})|\mathbf{A}^T=\mathbf{A}\}\) equipped with the operation \(\mathbf{A}*_G\mathbf{B}=\mathbf{A}\mathbf{B}\) forms a group.
- True
- False
MCQ 3
The set of symmetric invertible matrices together with usual matrix multiplication forms an abelian group.
- True
- False
MCQ 4
\((\mathbb N,+)\) is a subgroup of \((\mathbb Q,+).\)
- True
- False
MCQ 5
Let \(\phi : \mathrm{GL}(n,\mathbb{K}) \times M_{n,n}(\mathbb{K}) \to M_{n,n}(\mathbb{K}), \quad (\mathbf{C},\mathbf{A}) \mapsto \phi(\mathbf{C},\mathbf{A})=\mathbf{C}\mathbf{A}\mathbf{C}^{-1},\) then there is a diagonal matrix in the orbit \(\mathrm{GL}(n,\mathbb{K})\mathbf{A}\) for all \(\mathbf{A}\in M_{n,n}(\mathbb{K}).\)
- True
- False
MCQ 6
\((\mathbb R,\times)\) is group.
- True
- False
MCQ 7
The positive reals form a group with respect to multiplication.
- True
- False
MCQ 8
The negative reals form a group with respect to multiplication.
- True
- False
MCQ 9
The upper triangular matrices over a field form a group with respect to matrix multiplication.
- True
- False
MCQ 10
The strictly upper triangular matrices over a field form a group with respect to matrix multiplication.
- True
- False
MCQ 11
There exists a non-trivial group in which every element equals its own inverse.
- True
- False