Problem 8.8 Let n > 1 be a fixed natural number, and consider the n-fold cartesian product Z/2xxZ/2 = (Z/2)*". An element z € (Z/2)×" can be thought of as a string of bits of length n. For example, when n = 4, all elements of (Z/2)×4 are 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 (a) Compute the order of every element of (Z/2)*" for any n. (Try n = 1,2 first, then generalize.)
Problem 8.8 Let n > 1 be a fixed natural number, and consider the n-fold cartesian product Z/2xxZ/2 = (Z/2)*". An element z € (Z/2)×" can be thought of as a string of bits of length n. For example, when n = 4, all elements of (Z/2)×4 are 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 (a) Compute the order of every element of (Z/2)*" for any n. (Try n = 1,2 first, then generalize.)
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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Transcribed Image Text:Problem 8.8 Let n > 1 be a fixed natural number, and consider the
n-fold cartesian product
Z/2 x -..x Z/2 = (Z/2)*" .
An element z € (Z/2)*" can be thought of as a string of bits of
length n. For example, when n = 4, all elements of (Z/2)ת are
0000, 0001, 0010, 0011, 0100, 0101,0110, 0111
%3D
1000, 1001, 1010, 1011, 1100, 1101,1110, 1111
(a) Compute the order of every element of (Z/2)*" for any n. (Try
n = 1,2 first, then generalize.)
(b) Find all subgroups of (Z/2)*² =Z/2×Z/2.
(c) Find all subgroups of (Z/2)*³ = Z/2× Z/2×Z/2.
(d) (*) Let M be an n x n-matrix with entries in Z/2.
Show that the subset
{M ·x |x € (Z/2)*"}
is a subgroup.
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