Is 3 a generator for the group of multiplication modulo 7? Show why this is or...
Is 3 a generator for the group of multiplication modulo 7? Show why this is or is not the case.
Homework Answers
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Is 3 a generator for the group of multiplication modulo 7? Show
why this is or...
Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cycl...
Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cyclic subgroups of (G, ). (c) Explain why (G, ) is not isomorphic to the symmetric group S3. (d) State an isomorphism between (G, ) and (Z6, +).
abstract algebra 11. Modular arithmetic a. Write the table for modular addition (modulo 7) b. Write...
abstract algebra
11. Modular arithmetic a. Write the table for modular addition (modulo 7) b. Write the table for modular addition (modulo 8) 12. Modular multiplication a. Write the table for modular multiplication (modulo 7) b. Write the table for modular multiplication (modulo 8) c. Can you describe the difference you see in these cases?
Exercise 4.1: Explain/prove why the following sets and binary operations do not define groups (so just...
Exercise 4.1: Explain/prove why the following sets and binary operations do not define groups (so just try to determine one group axiom that fails to hold): 1) The set of polynomials of odd degree under addition. 2) The set of polynomials of odd degree under multiplication 3) The set of integers congruent to 1 modulo 11, under addition modulo 11. 4) The set of integers modulo 11, under multiplication modulo 11. 5) The set of nonzero integers modulo 4, under...
Let ne N. Show that Zn forn > 2 is not a group under multiplication as...
Let ne N. Show that Zn forn > 2 is not a group under multiplication as defined above. What happens for n = 1?
Let k > 3. Show that (1) 3 has order 2^(k-2) modulo 2^k . (2) {3,...
Let k > 3. Show that (1) 3 has order 2^(k-2) modulo 2^k . (2)
{3, -1} is a generating set for 2^k
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
Numbers 3,4,11 a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real...
Numbers 3,4,11
a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with integer entries e. exponentiation of integers 3. Which of the following binary operations are commutative? a. substraction of integers b. division of nonzero real numbers c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with real entries e. exponentiation of integers 4. Which of the following sets are closed...
Let k > 3. Show that (1) 3 has order 2^(k-2) modulo 2k . (2) {3, -1} is a generating set for 2k . Let k 3. Show that...
Let k > 3. Show that
(1) 3 has order 2^(k-2) modulo 2k .
(2) {3, -1} is a generating set for 2k .
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
use factors 7 and 6 to show an example of distributive property or multiplication
use factors 7 and 6 to show an example of distributive property or multiplication
3. The number 2 is a primitive root modulo 19; the powers of 2 modulo 19...
3. The number 2 is a primitive root modulo 19; the powers of 2 modulo 19 are listed below 21 22232425 26 228221212213214215216217 21 2 48 16 13 7 14918 175 36 125 101 Use this table to solve r 7 mod 19.
Modern Algebra 5) Consider the ollowing sets, S, together with the defined binary operation. In each...
Modern Algebra
5) Consider the ollowing sets, S, together with the defined binary operation. In each case, determine if the set is closed under the given operation, if the operation is associative and if the operation is commutative: ii) S R a -a b 6) Define the binary operation, multiplication modulo 3 in much the same way as we did addition modulo 3. That is, perform ordinary multiplication and then reduce the result modulo 3. Let S-(0, 1,2. Create two...
abstract algebra
11. Modular arithmetic a. Write the table for modular addition (modulo 7) b. Write the table for modular addition (modulo 8) 12. Modular multiplication a. Write the table for modular multiplication (modulo 7) b. Write the table for modular multiplication (modulo 8) c. Can you describe the difference you see in these cases?
Exercise 4.1: Explain/prove why the following sets and binary operations do not define groups (so just try to determine one group axiom that fails to hold): 1) The set of polynomials of odd degree under addition. 2) The set of polynomials of odd degree under multiplication 3) The set of integers congruent to 1 modulo 11, under addition modulo 11. 4) The set of integers modulo 11, under multiplication modulo 11. 5) The set of nonzero integers modulo 4, under...
Let ne N. Show that Zn forn > 2 is not a group under multiplication as defined above. What happens for n = 1?
Let k > 3. Show that (1) 3 has order 2^(k-2) modulo 2^k . (2)
{3, -1} is a generating set for 2^k
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
Numbers 3,4,11
a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with integer entries e. exponentiation of integers 3. Which of the following binary operations are commutative? a. substraction of integers b. division of nonzero real numbers c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with real entries e. exponentiation of integers 4. Which of the following sets are closed...
Let k > 3. Show that
(1) 3 has order 2^(k-2) modulo 2k .
(2) {3, -1} is a generating set for 2k .
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
Let k 3. Show that (1) 3 has order 2-2 modulo 2* (2) {3,- is a generating set for 2
3. The number 2 is a primitive root modulo 19; the powers of 2 modulo 19 are listed below 21 22232425 26 228221212213214215216217 21 2 48 16 13 7 14918 175 36 125 101 Use this table to solve r 7 mod 19.
Modern Algebra
5) Consider the ollowing sets, S, together with the defined binary operation. In each case, determine if the set is closed under the given operation, if the operation is associative and if the operation is commutative: ii) S R a -a b 6) Define the binary operation, multiplication modulo 3 in much the same way as we did addition modulo 3. That is, perform ordinary multiplication and then reduce the result modulo 3. Let S-(0, 1,2. Create two...
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