Let GL(2, 11) be the group of all invertible 2 x 2 matrices with entries in Z₁1, with group operation given my matrix multiplication. Consider the following two matrices in this group (where an entry listed as k is shorthand for [k]11): 3 10 A = (₁ 10), B = (3 ¹18). 1 8 (i) Show that A has order 5, B has order 2, and that BAB-¹ = A−¹.
Let GL(2, 11) be the group of all invertible 2 x 2 matrices with entries in Z₁1, with group operation given my matrix multiplication. Consider the following two matrices in this group (where an entry listed as k is shorthand for [k]11): 3 10 A = (₁ 10), B = (3 ¹18). 1 8 (i) Show that A has order 5, B has order 2, and that BAB-¹ = A−¹.
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 GL(2, 11) be the group of all invertible 2 × 2 matrices with entries in Z₁1,
with group operation given my matrix multiplication. Consider the following two
matrices in this group (where an entry listed as k is shorthand for [k]₁1):
A =
3
0
B =
3 10
8 8
Show that A has order 5, B has order 2, and that BAB-¹ = A-¹.
(ii) Consider the subset of GL(2,11) given by
G = {Am B : m, n € Z}.
Show that G is a subgroup of GL(2, 11).
(iii) List all the elements of G, together with their orders.
(iv) Identify G in terms of known groups.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86c8dcbb-d46d-4c91-a740-ef32ebf33ae0%2F7ae6e763-c8e6-4a04-882b-f57b689ce589%2Fxnyg8ea_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let GL(2, 11) be the group of all invertible 2 × 2 matrices with entries in Z₁1,
with group operation given my matrix multiplication. Consider the following two
matrices in this group (where an entry listed as k is shorthand for [k]₁1):
A =
3
0
B =
3 10
8 8
Show that A has order 5, B has order 2, and that BAB-¹ = A-¹.
(ii) Consider the subset of GL(2,11) given by
G = {Am B : m, n € Z}.
Show that G is a subgroup of GL(2, 11).
(iii) List all the elements of G, together with their orders.
(iv) Identify G in terms of known groups.
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