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Fundamental Mathmatics

1. a) State the Lagrange Theorem explaining any terms you use.
b) Let alpha, a member of S_11, be the permutation given by

alpha(1) = 7, alpha(2) = 5, alpha(3) = 1, alpha(4) = 2, alpha(5) = 8,
alpha(6) = 9, alpha(7) = 10, alpha(8) = 4, alpha(9) = 11, alpha(10) = 3, alpha(11) = 6.

Decompose the permutation alpha first as a product of disjoint cycles and then as a product of transpsitions. What are the order and sign of alpha and alpha^-1?

c) Show that if H and K are subgroups of a group G, then the intersect of H and K is also a subgroup of G. Show that if H and K have orders 9 and 8, respectively, then the intersect of H and K contains only one element.

2. State the First Isomorphism Theorem explaining any terms you use.
b) Let G = C* be the multiplicative group of nonzero complex numbers. Is the map f: G --> G a homomorphism, provided f is given by i) f(z) = iz, ii) f(z) = z^2, iii) f(z) = |z|, iv) f(z) = z-bar? Justify your answer.
c) Let K be a field and let G be the set of all matrices of the form
a b
0 c
where a, b, c is a member of K and a =/ 0, c =/ 0. Prove that G is a group under matrix multiplication. Prove that the map g: G --> K* x K* defined by
g * the matrix: a b = (a, c)
0 c
is a homomorphism. Here K* is the set of all non-zero elements of K, considered as a multiplicative group. Prove that the kernel of g is isomorphic to the additive group of the filed K. Deduce that the set H consisting of matrices of the form
1 b
0 1
is a normal subgroup of G and the quotient group G/H is isomorphic to K* xx K*. State clearly all results that you used.

3. a) Let V be a vector space over a field K and let f: V --> V be a linear map. Suppose v is a non-zero element of V and lambda is a member of K. Explain what it means to say that v is an eigenvector of f with eigenvalue lambda. Prove that V has a basis consisting of eigenvectors of f if and only if it has a basis with respect to which the matrix representing f is diagonal.
b) Let f: R --> R be a linear map given by
f * binomial (x y) = 1 -2 * (x y)
3 -1

Find the matrix A that corresponds to the mapping f in the basis
u_1 = (1 1), u_2 = (0 1).

c) Find the characteristic and minimal polynomials of the linear map f: R^3 --> R^3 given by the matrix
2 0 0
B = 1 0 1
1 -2 3

Is there a basis for R^3 for which the matrix of f is diagonal? Justify your answer.

4. a) Let V be an inner product spave and let g: V --> V be a map. Explain what it means to say that g is an isometry. Define what it means for a square matrix to be orthogonal. Prove that the product of two orthogonal matrices is orthogonal. Explain the relationship between orthogonal matrices and isometries.

b) Find a, b and c such that the matrix

1/3 0 a
2/3 1/sqrt(2) b
2/3 -1/sqrt(2) c

is orthogonal. Does this condition determine a, b and c uniquely?

c) Let V be a subspace of R defined by

V = {(x_1, x_2, x_3, x_4) is a member of R^4 | x_1 - x_2 + x_3 - x_4 = 0}

Find an orthonormal basis of V.

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1. (a) The Lagrange Theorem states that the order of any subgroup H of a finite group G divides the order of G.

(b) We are given the permutation of given by

We wish to write as the product of disjoint cycles and then as a product of transpositions. We also wish to determine the order and sign of and

To find the disjoint cycles in the product representation of , we iterate starting with a given index For we have and Thus we have found the cycle in the product representation of . Next we start with which is the smallest unused index. Now we have and Thus we have found the cycle Next we start with which is the smallest unused index. Now we have and Thus we have found the cycle Since this exhausts all indices, we have following product representation of :

To write as a product of transpositions, we note that the cycle may be written as Thus we have

Since is the product of eight transpositions, which is an even number, we see that is an even transposition, whence its sign is Since the inverse of every permutation has the same signature as the original permutation, we see that also has sign

Now the order of (and , which is necessarily the same) is equal to the least common multiple of the orders of each of the disjoint cycles in its product representation. Thus, the orders of and are both equal to

(c) We wish to show that if H and K are both subgroups of a group G, then is also a subgroup of G. We also wish to show that if H and K have orders 9 and 8 respectively, then has just one element.

To show that is a subgroup of G, it suffices to show that for every pair of elements a and b in , the element of G also lies in . Since H and K are both subgroups of G, and a and b each belong to both H and K, we see that belongs to both H and K, whence it belongs to . Thus we see that is a subgroup of G.

Now since is a group and is a subset of both H and K, it is a subgroup of both H and K. Thus, by Lagrange's Theorem, the order of must divide both the order of H and the order of K, and hence it must also the greatest common divisor of ...

Solution Summary

We solve various problems in abstract algebra, group theory and linear algebra in particular.

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