Mathematics 243, section 1 – Algebraic Structures Exam 3 – December 5, 2003

Directions

Do all work in the blue exam booklet. There are 100 possible regular points and 10 Extra Credit points.

I. Let G be a group, let a ∈ G be a fixed element, and consider the mapping h_{a} : G → G defined by h_{a}(x) = xa for all x ∈ G. A) (10) Show that h_{a} is one-to-one and onto. B) (10) If G is a finite group (that is, o(G) is finite), what does the result from part A

say about the group operation table of G?

II. A) (20) In the group G = Z_{24}, in which the operation is addition mod 24, find all of the

generators of the cyclic subgroup H = ⟨[15]⟩. B) (20) Write the element σ = (1532)(24368)(12479) in the permutation group S_{9} as a

product of disjoint cycles and determine the order of σ.

III. Let ϕ : G → H be a group homomorphism. A) (10) Show that if e_{G} and e_{H} are the identity elements in the groups G and H, then

ϕ(e_{G}) = e_{H}. B) (10) Show that ker(ϕ) is a subgroup of G.

IV. Let G be the set of all 2 × 2 matrices with real entries of the form A = ( \begin{matrix}a & b \\ −b & a \end{matrix} ),

where a, b are any real numbers, at least one of which is not zero. A) (10) Is G closed under the operation of matrix multiplication? B) (10) Does every element of G have a (multiplicative) inverse which is an element of

G? Note: We discussed a general rule for finding the inverse of a 2 × 2 matrix in class. If you don’t remember it, you can “buy” it from me during the exam in return for a reduction of 5 points on your score for this problem.

Extra Credit (10) Give an example of an invertible 2 × 2 matrix that has order 7 as an element of the group GL_{2}(R).

The General Pattern for 2 × 2 Matrix Inverses If

A = ( \begin{matrix}a & b \\ c & d \end{matrix} )

then A has a inverse matrix (for multiplication) if and only if det(A) = ad − bc ≠ 0, and then

A^{−1} = ( \begin{matrix}\frac{d}{ad−bc} & \frac{−b}{ad−bc} \\ \frac{−c}{ad−bc} & \frac{a}{ad−bc} \end{matrix} )