# Groups of Order 18

Construct two non-isomorphic, nonabelian groups of order 18.

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There are three known, different non-Abelian groups of order 18. There are the ...

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Two nonisomorphic, non-abelian groups of order 18 are constructed.

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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