Please write your answer clearly?
Please write your answer clearly? 4. Let K<HCG. Suppose G:H] =m and H:K] =n. Prove that...
Please do all parts and write your answer clearly. thanks! 6. a. Prove that there is a finite field E with 729 elements and provide the quotient ring that is isomorphic to E. b. Draw the lattice of subfields for E.
Please provide the theorems and definitions you use. 1. Let K be a subgroup of a group G. Let T denote the set of all distinct right cosets of K in G and A(T) be the permutation group of T. Prove the following statements. (a) For each a EG, the function fa:T T given by fa(Kb) = Kba is a bijection. (b) The function : G + A(T) given by pla) = fa-1 is a group homomorphism whose kernel is...
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...
Let M be a DFA that recognizes a finite language A, and suppose M has n states. Determine if the following statement is true or false: if w Element of A, then |w| < = n. Prove your answer.
Let U ? Rmxn. Prove that if UTI-In, then n < m.
k=42, m=18 n=4 11. Let F:R → R be a function such that (t+m)(n+1) (n+ m F(t) = for t <-m, f or-m <t<n. for n<t<k, for t > k. nA - 1 Find A and B knowing that F is the cumulative distribution function of a random variable X such that P(X = k) = . Please provide only the value of parameter B in the space specified below. ANSWER: B= Solution:
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G. (a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then...
please solve it very clear with explain your answer abstract algebra 2. Suppose that G and G' are groups and N G, N' 4G'. Prove that (G x G')/(N x N') = G/N x G'/N' <H<G with N 4G. Prove that H/N is a subgroup of 4. Suppose that G is a group, N G/N.
4. Here is a fact about permutations: (*) nPr= n!/(n-k)!, for all k =n. Let's prove this via mathematical induction for the fixed case k-3. 2 of 3 (i) Write clearly the statement (**) we wish to prove. Be sure your statement includes the phrase "for all n" (ii) State explicitly the assumption in (*) we will thus automatically make about k-2 (ii) Now recall that to prove by induction means to show that IfmPm!/lm-k)! is true for all km...