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 in G is also in N?

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?