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.

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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 15E: 15. Repeat Exercise with, the multiplicative group of matrices in Exercise of Section. 14. Let be...

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