Homework Answers
Please mention what is C_1 and
C=2
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi...
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi-1 Gi means Gi-1 is a normal subgroup of Gi. For example, any abelian group is solvable: If G s abelian, take Go f1), Gi- G. Then G1/Go G is abelian and hence G is solvable (a) Show that S3 is solvable Suggestion: Let Go- [l),Gı-(123)), and G2 -G. Here (123)) is the subgroup generated by the 3-cycle...
Let be a permutation of {1,2,……n}.Let -1 be the (n-1)-tuple with one element from missing. Alice...
Let
be a permutation of {1,2,……n}.Let
-1 be the (n-1)-tuple with one element from
missing.
Alice shows Bob
-1[i] one by one in the increasing order of i from 1 to
(n-1).bob’s task is to compute the missing element from
-1 that is in
with very limited – O(log n) bits – of memory.
Design an algorithm to compute the missing element in this
memory-limited and access-limited model, i.e Alice can only show
each number to Bob once, and Bob...
5. Product groups: (a) Let G and G' be groups. Explain how one turns G ×...
5. Product groups: (a) Let G and G' be groups. Explain how one turns G × G, into a group by defining multiplication and identifying inverses and the identity element. (b) If G is an abelian group of order 30 and GZm x Zn how may possibilities are there for the whole numbers m and n? (Assume m S n for clarity and list the possibilities)
7. Let A be a Abelian group of finite order n and let m be a...
7. Let A be a Abelian group of finite order n and let m be a natural number. Define a map Om : A + A by Om(a) = a". Prove that Om is a homomorphism of A and identify the kernel of øm. Determine when om is an isomorphism.
3. Let M be a manifold and let G C Homeo(M) be a group acting on M. Suppose that this group actio...
3. Let M be a manifold and let G C Homeo(M) be a group acting on M. Suppose that this group action is properly discontinuous and free prove that the quotient space M/G is a manifold. For this problem properly discontinuous means that if K c M is compact then the set {ge G | g(K) n/Kメ0) is finite) and free means the only element of g that fixes any point of M is the identity.
3. Let M be...
1. Show that the set of rational numbers of the form m /n, where m, n...
1. Show that the set of rational numbers of the form m /n, where m, n E Z and n is odd is a subgroup of QQ under addition. 2. Let H, K be subgroups of a group G. Prove: H n K is a subgroup of G 3. Let G be an abelian group. Let S-aEG o(a) is finite . Show that S is a subgroup of G 4. What is the largest order of a permutation in S10?...
Problem 4. Let G be a group. Recall that the order of an element g G...
Problem 4. Let G be a group. Recall that the order of an element g G is the smallest k such that gk = 1 (or 00, if such a k doesn't exist). (a) Find the order of each element of the symmetric group S (b) Let σ-(135)(24) and τ-(15)(23)(4) be permutations in S5. Find the cycle decompositions for (c) Let σ-(123456789). Compute ơ-i, σ3, σ-50, and σί006 (d) Find all numbers n such that Ss contains an element of...
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.) a 4.7. Let n 2 3. Every element...
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.) a 4.7. Let n 2 3. Every element of An can be written as the product of 3-cycles.
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.)
a 4.7. Let n 2 3. Every element of An can be written as the product of 3-cycles.
3. (10%) Let C = AB, where A and B are both n by n matrices....
3. (10%) Let C = AB, where A and B are both n by n matrices. The element located at row i and column of C is represented by C, and computed as C, = A, B, + A,B2, + ... + 4,B, a) Express C, using the X (summation) notation. b) Evaluate c, if Ak = 2 and B, = 3 for all k, k = 1,2,...n.
Let M be a 8:27 AM right R-module, N be an (R,T)-bimodule, and L be a...
Let M be a 8:27 AM right R-module, N be an (R,T)-bimodule, and L be a left T-module. Let e: (MN)* L M R (NB, L) be given by e (moon, e) = m (nol). Let m.con, mone MORN, and lEl. Prove e (lm, BR.) + (m₂ Ore), d)= e(m, on, d) + (mon, e). This is the proof I'm working on. I need to show the map I've defined (and which is defined towards the middle of the proof)...
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi-1 Gi means Gi-1 is a normal subgroup of Gi. For example, any abelian group is solvable: If G s abelian, take Go f1), Gi- G. Then G1/Go G is abelian and hence G is solvable (a) Show that S3 is solvable Suggestion: Let Go- [l),Gı-(123)), and G2 -G. Here (123)) is the subgroup generated by the 3-cycle...
Let
be a permutation of {1,2,……n}.Let
-1 be the (n-1)-tuple with one element from
missing.
Alice shows Bob
-1[i] one by one in the increasing order of i from 1 to
(n-1).bob’s task is to compute the missing element from
-1 that is in
with very limited – O(log n) bits – of memory.
Design an algorithm to compute the missing element in this
memory-limited and access-limited model, i.e Alice can only show
each number to Bob once, and Bob...
5. Product groups: (a) Let G and G' be groups. Explain how one turns G × G, into a group by defining multiplication and identifying inverses and the identity element. (b) If G is an abelian group of order 30 and GZm x Zn how may possibilities are there for the whole numbers m and n? (Assume m S n for clarity and list the possibilities)
7. Let A be a Abelian group of finite order n and let m be a natural number. Define a map Om : A + A by Om(a) = a". Prove that Om is a homomorphism of A and identify the kernel of øm. Determine when om is an isomorphism.
3. Let M be a manifold and let G C Homeo(M) be a group acting on M. Suppose that this group action is properly discontinuous and free prove that the quotient space M/G is a manifold. For this problem properly discontinuous means that if K c M is compact then the set {ge G | g(K) n/Kメ0) is finite) and free means the only element of g that fixes any point of M is the identity.
3. Let M be...
1. Show that the set of rational numbers of the form m /n, where m, n E Z and n is odd is a subgroup of QQ under addition. 2. Let H, K be subgroups of a group G. Prove: H n K is a subgroup of G 3. Let G be an abelian group. Let S-aEG o(a) is finite . Show that S is a subgroup of G 4. What is the largest order of a permutation in S10?...
Problem 4. Let G be a group. Recall that the order of an element g G is the smallest k such that gk = 1 (or 00, if such a k doesn't exist). (a) Find the order of each element of the symmetric group S (b) Let σ-(135)(24) and τ-(15)(23)(4) be permutations in S5. Find the cycle decompositions for (c) Let σ-(123456789). Compute ơ-i, σ3, σ-50, and σί006 (d) Find all numbers n such that Ss contains an element of...
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.) a 4.7. Let n 2 3. Every element of An can be written as the product of 3-cycles.
15. If φ: Sn Sn is a group homomorphism, prove that φ(An) c An. (Hint: Use Lemma 4.7.)
a 4.7. Let n 2 3. Every element of An can be written as the product of 3-cycles.
3. (10%) Let C = AB, where A and B are both n by n matrices. The element located at row i and column of C is represented by C, and computed as C, = A, B, + A,B2, + ... + 4,B, a) Express C, using the X (summation) notation. b) Evaluate c, if Ak = 2 and B, = 3 for all k, k = 1,2,...n.
Let M be a 8:27 AM right R-module, N be an (R,T)-bimodule, and L be a left T-module. Let e: (MN)* L M R (NB, L) be given by e (moon, e) = m (nol). Let m.con, mone MORN, and lEl. Prove e (lm, BR.) + (m₂ Ore), d)= e(m, on, d) + (mon, e). This is the proof I'm working on. I need to show the map I've defined (and which is defined towards the middle of the proof)...
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