Group Theory Questions

Group Theory Questions. See attached file for full problem description.

Mobius Functions, Euler Functions and Carmicheal's Conjecture

1) Prove that in is a positive integer. ( : is the Mobius function) Hint: one of the four argument is divisible by 4. 2) If is a prime and . Show that ( : is the Euler function) 3) a. Prove that is an integer if n is a prime and that it is not an integer b. Prove that is not an integer if n is divisi ...continues

Ring Ideal <2,x> in Z[x]

Let I be the ideal <2,x> in Z[x] where Z[x] is the Ring of Polynomials in Z and <2,x> is of the form 2k+(a_1)(x_1)+...+(a_n)(x_n). How many elements can Z[x]/I have?

Let a commutative ring R be generated by {a_1, a_2, ..., a_n}

Let a commutative ring R be generated by {a_1, a_2, ..., a_n} such that [a_1, a_2, ... , a_n] = {(a_1xr_1) + (a_2xr_2) + ... + (a_nxr_n) for r_1, ..., r_n in set of Reals}. I need to show this set is an ideal. Do I just need to show that it satisfies the commutative properties of the ideal?

Automorphisms

Show that all automorphisms of a group G form a group under function composition. Then show that the inner automorphisms of G, defined by f : G--->G so that f(x) = (a^(-1))(x)(a), form a normal subgroup of the group of all automorphisms. For the first part, I can see that we need to show that f(g(x)) = g(f(x)) for x in ...continues

Direct Products, Permutations and Orders of Elements

Let a be the permutation (1 2 3) in A_4. What is the order of the element (3, 7, a) in the group U(10) direct product Z_42 direct product A_4.

Isomorphisms, Residue Classes and Multiplication Tables

Show that U(10) is isomorphic to Z_4 and write out the isomorphism explicitly. I know that U(10) and Z_4 are both cyclic, thus they are ismorphic but for writing out the isomorphism, I need assistance.

Does the set {irrational numbers} U {1} form a group under multiplication?

Does the set {irrational numbers} U {1} form a group under multiplication? Either show this or explain why it is not true.

Multiplication Table for U(12); Elements in the Subgroup <5>; Is the group U(12) cyclic?

What is the complete multiplication table for U(12)? What are the elements in the subgroup <5>? Is the group U(12) cyclic?

Direct Products and Isomorphisms

Let G = Z_3 direct product Z_3 direct product Z_3 and let H be the subgroup of SL(3, Z_3) consisting of 1 a b the matrix H = { 0 1 c with a, b, c in Z_3 } 0 0 1 What is the order of G and H and are G and H isomorphic?