Discussion
Let G be the group of invertible \(2 \times 2\) matrices over the field \(\mathbf{Z}_{2} = \{0, 1\}\). Then the number of elements in G is :
Answer: A
The group GL(2, \(\mathbf{Z}_{2}\)) has 6 elements.
Choose the first column as any non-zero vector in \(\mathbf{Z}_{2}^{2}\): \(4 - 1 = 3\) ways. The second column must not be a multiple of the first, which rules out 0 and the first column itself: \(4 - 2 = 2\) ways. So
\[ \begin{aligned} |GL(2, \mathbf{Z}_{2})| &= (2^{2} - 1)(2^{2} - 2) \\ &= 3 \cdot 2 = 6 \end{aligned} \]
There are \(2^{4} = 16\) matrices in all, and 10 of them are singular, leaving 6. The options 2, 4 and 8 undercount; the group is in fact isomorphic to \(S_{3}\), acting on the three non-zero vectors.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development