Normal Subgroups

Let G be a finite group, let N be a normal subgroup of G, and let x be an element of G. Show that if the order of x in G is relatively prime to |G|/|N|, then x is an element of N. We know that xNx^(-1) is identical to N when N is normal, for any x. Also we know that |G|/|N| is a factor of (or divides) |G|. How to show x i ...continues

Group Theory Questions

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

Mobious, Euler, Carmicheal - Algebra

Mobious, Euler, Carmicheal - Algebra. See attached file for full problem description.

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?

Show that all automorphisms of a group G form a group under function composition.

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 product problem

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.

Isomorphism Problem

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.

Group question

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

Is 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?