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Problem 8.8 Let n > 1 be a fixed natural number, and consider the n-fold cartesian product Z/2 × -..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)*ª are %3D 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/2 × -..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)*ª are %3D 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
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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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 r € (Z/2)*n 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.) (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.
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 r € (Z/2)*n 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.) (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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