### Isomorphisms and automorphisms.

This week lecture is taught about Isomorphism, automorphism and Inner automorphism, but I don't understand what they are. Can you give some simple examples?

This week lecture is taught about Isomorphism, automorphism and Inner automorphism, but I don't understand what they are. Can you give some simple examples?

Please help. I only need answers with brief explanations. No need of detailed working. (See attached file for full problem description) --- State whether the following are true or false with reasons: 1. If a in S6, then an =1 for some n greater than or equal to 1. 2. If axa-1=bxb-1, then a=b 3. The function e

Suppose that T is a linear operator on a two dimensional vector space V, and that T is not multiple of the identity transformation. Show that T has a cyclic vector (i.e., there exists v V such that {v, Tv } is a base for V). Please see the attached file for the fully formatted problems.

Show that a homomorphism from a field onto a ring with more than one element must be an isomorphism. Recall The the function f is an isomorphism if and only if f is onto and Kernel ={ 0}. Please explain step by step with reasons in every step.

Give the order and describe a generator of the group G(GF(729)/ GF(9)).

Modern Algebra Group Theory (LXVI) Automorphism of a Group Let G be a g

Modern Algebra Group Theory (LXI) Automorphism of a Group Is the mapping given below an automorphism of the group ? G group of integers under addition, T:x --> -x

(See attached file for full problem description with proper symbols) --- a) Under what conditions will two path classes, and , from to , give rise to the same isomorphism of onto ? b) Let be an arcwise-connected space. Under what conditions is the following true: For any two points , all path classes from

Group theory Modern Algebra Group Theory (LV)

Let and be Banach spaces and let . (Note: is a set of all continuous linear transformations ). Show that there is a constant such that for iff and gra A is closed. ---

Consider the M/M/s queue, with arrival rate... 5. Consider the M/M/s queue, with arrival rate Λ >0 and service rate μ >0. (a) Find the condition involving Λ, μ and s that is necessary for there to exist a stationary distribution. Why does this condition makes sense? (b) Find the long-run proportion of time that there ar

We are working on the proof of showing G (the group of rigid motions of a regular dodecahedron) is isomorphic to the alternating group A_5. Lemma: Let H be a normal subgroup of a finite group G, and let x be an element of G. If o(x) and [G:H] are relatively prime, then x is in H. Theorem: Any 60 element group having 24 el

In each of Exercises 40 through 46 following, a linear space V is given and a mapping T : V→V is defined as indicated. In each case determine whether T is a linear mapping. If T is linear, determine the kernel (or null space) and range, and compute the dimension of each of these subspaces wherever they are finite-dimension

For odd m>=3, prove that D2m is isomorphic to Dm x Z2. Where D is the Dihedral group and Z is the group of integers.

What is the best statement that you can make about the existence and uniqueness of the solution of the following initial value problems? (a) y'= sin(ty)+1/t, y(1)=2 (See attachment for full question)

Let V = Q3 and let ' be the linear transformation from V to itself: '(x, y, z) = (9x + 4y + 5z,−4x − 3z,−6x − 4y − 2z), x, y, z E Q With respect to the standard basis B find the matrix representing this linear transformation. Take the basis E = {(2,−1,−2), (1, 0,−1), (3,−2

1) Define a linear transformation.... a) Find a basis for Ker T. b) Find a basis for Im T. c) Is T an onto map? d) Is T a one-to-one map? 2) Define a linear transformation... a) Find the matrix for T with respect to the standard basis. b) Find the matrix for T with respect to { ( ) , ( ) , ( ) } as the basis for R and t

I need help and an explanation for the following: Using the matrix A = 1 -1 0 0 -1 1 -1 2 -1 to compute TsubA(x), for x = (1, 2, 3)^T. Here TsubA: R^3 into R^3 is defined by TsubA(x) = Ax. Also describe the kernel of the transformation TsubA (that is state what a typical vector in ker T looks like).

Help with this linear transformation problem. Please help with the following problem. Provide step by step calculations. A. What is the order of the element 14 + <8> in the factor group Z24 / <8>? I know that if I let G = Z24 and H = <8> , then H = <8> = {0,8,16}. So Z24/<8> = {0+H,1+H,2+H,3+H,4+H,5+H,6+H,7+H}.

Determine whether the mapping phi: Z -> Z which is defined by phi(n) = n + 1 is an isomorphism from the binary structure (Z, +) to the binary structure (Z, +). If not, explain why and give a counter-example. See the attached file.

Let ... be a homomorphism of a group ... into a group... . Show that if is the identity element of... , then is the identity element... in ... . Please see the attached file for the fully formatted problems.

Prove that mapping... is a homomorphism. Note: both groups are under addition. Please see attached for full question.

Given an onto ring homomorphism Φ: R1 --> R2 we saw in class that if I <u>C</u> R1 is an ideal containing the kernel of Φ, then Φ(I) is an ideal of R2. In addition, we saw that if J <u>C</u> R2 is an ideal, then Φˉ¹(J) is an ideal of R1 containing kernel of Φ. (i) Show that if I <u>C</u> R1 is a

Suppose you have N jobs that have to be processed on a single machine. For i = 1, 2, . . . ,N, job i requires pi units of time on the machine, and has weight wi. The objective is to schedule these jobs so as to minimize the sum of the weighted completion time of all the jobs, where the completion time of job i is the time at w

Let T(z) = (az+b)/(cz+d), where ad-bc≠0, be any linear fractional transformation other than T(z) = z. Show that T^-1 = T only if d = -a. See the attached file.

1. Find the linear transformation that maps the points z_1 = 2, z_2 = i, z_3 = -2 onto the points w_1 = 1, w_2 = i, w_3 = -1. Ans. w = (3z + 2i)/(iz + 6)

3. Find the region onto which the half plane y > 0 is mapped by the transformation w = (1+ i)z by using (a) polar coordinates; (b) rectangular coordinates. Sketch the region. Ans. v > u. (This problem is from linear transformations.) (Please try to draw the graph clearly).

From the cubics P3 to the fourth degree polynomials P4, what matrix represents multiplication by 2 + 3t? The columns of the 5 by 4 matrix A come from applying the transformation to each basis vector x1 = 1, x2 = t, x3 = t2, x4 = t3. Verify directly from c2 + s2 = 1 that the reflection matrices satisfy H2 = I. c = cosine s

Derive the equation of the line through the points ... and ... in the ... plane, shown in Fig. 37. Then use it to find the linear function ... which can be used in equation (9), Sec. 38. to transform representation (2) in that section into representation (10) there.... The parametric representation used For any given arc C is

At the start of the year a company wants to invest excess cash in one-month, three-month and six-month CD's. The company is somewhat conservative and wants to make sure it has a safety margin of cash on hand each month. (left over from previous month/ available at the outset, plus principal and interest from CD's that have becom