2. Let n be a positive integer greater than 1. An element r in Z, is called nilpotent if x* = 0 for some positive integer k. For example, 0 and 2 are nilpotent in Z4, and 0, 3 and 6 are nilpotent in Zg. Let N = N(Z„) denote the set of all nilpotent elements in Z, and put 1+N = {1+z | z € N} . (a) Prove that N is closed under addition and multiplication. (b) Verify that N is an abelian group with respect to addition. (c) Prove that N is a group with respect to multiplication if and only if n is not divisible by the square of a prime number, in which case the group N is trivial.

2. Let n be a positive integer greater than 1. An element r in Z, is called nilpotent if x* = 0 for some positive integer k. For example, 0 and 2 are nilpotent in Z4, and 0, 3 and 6 are nilpotent in Zg. Let N = N(Z„) denote the set of all nilpotent elements in Z, and put 1+N = {1+z | z € N} . (a) Prove that N is closed under addition and multiplication. (b) Verify that N is an abelian group with respect to addition. (c) Prove that N is a group with respect to multiplication if and only if n is not divisible by the square of a prime number, in which case the group N is trivial.

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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Question

2. Let n be a positive integer greater than 1.
An element z in Z, is called nilpotent if a* = 0 for some positive integer k.
For example, 0 and 2 are nilpotent in Z4, and 0, 3 and 6 are nilpotent in Zg.
Let N = N(Zn) denote the set of all nilpotent elements in Z, and put
1+N = {1+z|z E N}.
(a) Prove that N is closed under addition and multiplication.
(b) Verify that N is an abelian group with respect to addition.
(c) Prove that N is a group with respect to multiplication if and only if n is
not divisible by the square of a prime number, in which case the group N is
trivial.
(d) Prove that 1+ N is a group with respect to multiplication.
(e) Verify that 1+ N(Z27) is a cyclic group under multiplication, but 1+N(Z16)
is not a cyclic group under multiplication.

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