Let H = 〈10〉 , N = 〈4〉 in Z40 . (a) List the elements in...
Let H = 〈10〉 , N = 〈4〉 in Z40 .
(a) List the elements in H N (or you might say H + N in this case) and list the elements in H ∩ N .
(b) List the cosets in HN / N , showing the elements in each coset.
(c) List the cosets in H / ( H∩N ) , showing the elements in each coset.
(d) Give the correspondence between HN / N and H / ( H∩N ) , as described in the proof of the Second Isomorphism Theorem.
Homework Answers
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Answer Question 5 . Name: 1. Prove that if N is a subgroup of index 2...
Answer Question 5 .
Name: 1. Prove that if N is a subgroup of index 2 in a group G, then N is normal in G 2. Let N < SI consists of all those permutations ơ such that o(4)-4. Is N nonnal in sa? 3. Let G be a finite group and H a subgroup of G of order . If H is the only subgroup of G of order n, then is normal in G 4. Let G...
Can you explain these for me..I mean give example for each of them to get the...
Can you explain these for me..I mean give example for
each of them to get the idea
THE FIRST ISOMORPHISM THEOREM Let: G M be a homomorphism with Ker() = K, and Im() = I. Then there is a natural isomorphism 0:1 ™G/K which is surjective. Thus, I G/K. THE SECOND ISOMORPHISM THEOREM Suppose the N is a normal subgroup of G, and that H is a subgroup of G. Then H/( HN) = (HN)/N. THE THIRD ISOMORPHISM THEOREM Let...
10. Let G = D. be the dihedral group on the octagon and let N =...
10. Let G = D. be the dihedral group on the octagon and let N = (r) be the subgroup of G generated by r4. (a) Prove that N is a normal subgroup of G. (b) If G =D3/N, find G. (c) Using the bar notation for cosets, show that G = {e, F, 2, 3, 5, 87, 82, 83}. Hint: Show that the RHS consists of distinct elements and then use part (b). (d) Prove that G-D4. Hint: It...
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,):...
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
n=24 k=4 5. If the third digit in your student ID is odd, let n =...
n=24
k=4
5. If the third digit in your student ID is odd, let n = 24. Otherwise, let n = 12. If the 5th digit in your student ID is odd, let k = 4. Otherwise, let k = 3. Give an example of a nonabelian group G of order n and a subgroup H of order k. Then list all of the cosets of G/H.
4 Let G be an unknown group of order 8. By the First Sylow Theorem, G must contain a subgroup H of order 4 (a) If al...
4 Let G be an unknown group of order 8. By the First Sylow Theorem, G must contain a subgroup H of order 4 (a) If all subgroups of G of order 4 are isomorphic to V then what group must G be? Completely justify your answer. (b) Next, suppose that G has a subgroup H one of the following C Then G has a Cayley diagram like Find all possibilities for finishing the Cayley diagram. (c) Label each completed...
Let G- be a generator matrix for a block code (not necessarily a "good" code) a) b) c) What is th...
Let G- be a generator matrix for a block code (not necessarily a "good" code) a) b) c) What is the n, k, the rate and the bandwidth expansion for this code? Find the parity check matrix H )Build the standard array for the code. Assume the coset leaders are vectors with one "l", starting from the left side of the vector, i.e., the first coset leader will be (1 0...), the second (01 0 ...) starting again from the...
(4) This exercise outlines a proof that [21 KI 1//IIKİ whenever H and K are subgroups of a group ...
(4) This exercise outlines a proof that [21 KI 1//IIKİ whenever H and K are subgroups of a group G. (Note that HK-{hk | he H and k E K). The set HK is not always a subgroup of G.) Let -{hK | h є H). Define an action . . H x Ο Ο by the rule hị . ћК hihi. (You may assume that this is an action.) (a) Prove that OH(X). (b) Prove that HK-Hn K. (Here...
3. Let X be a set of n elements to be maintained as a linked list:...
3. Let X be a set of n elements to be maintained as a linked list: Assume that the cost of examining a particular element Xi is Ci. Note that to examine Xi, one needs to scan through all elements in front of Xi. Let Pi be the probability of searching for element Xi, so the total cost for all searches is (a) Give a counterexample to show that storing elements in non-increasing order of Pi does not necessarily minimize...
4. List all left cosets of An in Sn. (See 3.7.11.) For a given permutation o...
4. List all left cosets of An in Sn. (See 3.7.11.) For a given permutation o in Sn, how can you tell from o which coset o An is? Example 3.7.11. Pick a positive integer n > 2 and consider the group S. We define An = {o ESO is an even permutation). We will use the first theorem above to verify that An is a subgroup of S First of all, the identity is defined to be an even...
Answer Question 5 .
Name: 1. Prove that if N is a subgroup of index 2 in a group G, then N is normal in G 2. Let N < SI consists of all those permutations ơ such that o(4)-4. Is N nonnal in sa? 3. Let G be a finite group and H a subgroup of G of order . If H is the only subgroup of G of order n, then is normal in G 4. Let G...
Can you explain these for me..I mean give example for
each of them to get the idea
THE FIRST ISOMORPHISM THEOREM Let: G M be a homomorphism with Ker() = K, and Im() = I. Then there is a natural isomorphism 0:1 ™G/K which is surjective. Thus, I G/K. THE SECOND ISOMORPHISM THEOREM Suppose the N is a normal subgroup of G, and that H is a subgroup of G. Then H/( HN) = (HN)/N. THE THIRD ISOMORPHISM THEOREM Let...
10. Let G = D. be the dihedral group on the octagon and let N = (r) be the subgroup of G generated by r4. (a) Prove that N is a normal subgroup of G. (b) If G =D3/N, find G. (c) Using the bar notation for cosets, show that G = {e, F, 2, 3, 5, 87, 82, 83}. Hint: Show that the RHS consists of distinct elements and then use part (b). (d) Prove that G-D4. Hint: It...
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
n=24
k=4
5. If the third digit in your student ID is odd, let n = 24. Otherwise, let n = 12. If the 5th digit in your student ID is odd, let k = 4. Otherwise, let k = 3. Give an example of a nonabelian group G of order n and a subgroup H of order k. Then list all of the cosets of G/H.
4 Let G be an unknown group of order 8. By the First Sylow Theorem, G must contain a subgroup H of order 4 (a) If all subgroups of G of order 4 are isomorphic to V then what group must G be? Completely justify your answer. (b) Next, suppose that G has a subgroup H one of the following C Then G has a Cayley diagram like Find all possibilities for finishing the Cayley diagram. (c) Label each completed...
Let G- be a generator matrix for a block code (not necessarily a "good" code) a) b) c) What is the n, k, the rate and the bandwidth expansion for this code? Find the parity check matrix H )Build the standard array for the code. Assume the coset leaders are vectors with one "l", starting from the left side of the vector, i.e., the first coset leader will be (1 0...), the second (01 0 ...) starting again from the...
(4) This exercise outlines a proof that [21 KI 1//IIKİ whenever H and K are subgroups of a group G. (Note that HK-{hk | he H and k E K). The set HK is not always a subgroup of G.) Let -{hK | h є H). Define an action . . H x Ο Ο by the rule hị . ћК hihi. (You may assume that this is an action.) (a) Prove that OH(X). (b) Prove that HK-Hn K. (Here...
3. Let X be a set of n elements to be maintained as a linked list: Assume that the cost of examining a particular element Xi is Ci. Note that to examine Xi, one needs to scan through all elements in front of Xi. Let Pi be the probability of searching for element Xi, so the total cost for all searches is (a) Give a counterexample to show that storing elements in non-increasing order of Pi does not necessarily minimize...
4. List all left cosets of An in Sn. (See 3.7.11.) For a given permutation o in Sn, how can you tell from o which coset o An is? Example 3.7.11. Pick a positive integer n > 2 and consider the group S. We define An = {o ESO is an even permutation). We will use the first theorem above to verify that An is a subgroup of S First of all, the identity is defined to be an even...
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