## Dr. Sarah's Problem Set 6

Problem 1 Give two formal reasons why 2Z+1 (odd integers) are not a group under integer addition.

Problem 2 Use DeMoivre's Thm from the Last Problem Set to prove that Theta_n={cos 2pik/n +i sin 2pik/n | k=0,1,2,...,n-1} (the complex zeros of x^n-1) is closed under multiplication. Prove that 3 and 4 in the definition of a group holds for Theta_n. Since complex multiplication is associative, this proves that Theta_n is a group under complex multiplication.

Problem 3 Prove that G={A in GL(2,R) | det A is non-zero} is a non-abelian group under matrix multiplication. Prove that G is not a group under matrix addition.

Problem 4 An abstract algebra teacher in the 1970s intended to give a typist a list of 9 integers that form a group under multiplication modulo 91. Instead, one of the integers was inadvertently left out so that the list appeared as
1, 9, 16, 22, 53, 74, 79, 81
Which integer was left out? After finding the integer, prove that your collection of numbers is a group under the operation multiplication modulo 91. What is the order of this group? Why? Prove that your collection of numbers is not a group under the operation addition modulo 91.

The set of all positive integers less than n and relatively prime (no commong divisors) to n is a group under multiplication modulo n. Explain why the set of all positive integers less than 91 and relatively prime to 91 has 72 elements. Call this group Mod91group.

Lagrange's Theorem says that if G is a finite group and H is a subgroup of G, then the order of H divides the order of G. Could you have used Lagrange's Theorem to prove that
1, 9, 16, 22, 53, 74, 79, 81
could not be a subgroup of Mod91group? Explain? Could you use Lagrange's Theorem to prove that
1, 9, 16, 22, 74, 79, 81
can not be a subgroup of Mod91group? Explain?

Problem 5 Molecules with chemical formulas of the form A B_4, such as methane (C H_4) and carbon tetrachloride (C CL_4), have symmetry group the rotations of a tetrahedron, call it T. Build a 3-d model of a tetrahedron (be creative!) and label the vertices with colors or some other labeling system. Describe in detail the 12 rotations in T of the tetrahedron that rotate the tetrahedron back onto itself in our 3-space. Pick one of your non-identity rotations and call it r. Prove that r composed with s is in T for all s in T, where composition is composition of functions. Be sure to explain your compositions in detail.

Convince yourself (but do not write down the proof) that T is a group under composition.