1 0 0 Let G be the set of all 3 × 3 matrices of the form 1 0 a 1 (a) Show that if a, b, c E Z3, then G is a group of exponent 3. (b) Show that if a, b, c E Z2, then G is a group of exponent 4.
1 0 0 Let G be the set of all 3 × 3 matrices of the form 1 0 a 1 (a) Show that if a, b, c E Z3, then G is a group of exponent 3. (b) Show that if a, b, c E Z2, then G is a group of exponent 4.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let \( G \) be the set of all \( 3 \times 3 \) matrices of the form
\[
\begin{bmatrix}
1 & 0 & 0 \\
a & 1 & 0 \\
b & c & 1
\end{bmatrix}
\].
(a) Show that if \( a, b, c \in \mathbb{Z}_3 \), then \( G \) is a group of exponent 3.
(b) Show that if \( a, b, c \in \mathbb{Z}_2 \), then \( G \) is a group of exponent 4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2Fdaad6fa2-bc05-4ec3-abe2-0264ae53ba47%2F7ghsgz_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( G \) be the set of all \( 3 \times 3 \) matrices of the form
\[
\begin{bmatrix}
1 & 0 & 0 \\
a & 1 & 0 \\
b & c & 1
\end{bmatrix}
\].
(a) Show that if \( a, b, c \in \mathbb{Z}_3 \), then \( G \) is a group of exponent 3.
(b) Show that if \( a, b, c \in \mathbb{Z}_2 \), then \( G \) is a group of exponent 4.
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