Abstract Algebra Problems

1. Let G ne a finite group and N be a normal subgroup of G. Let [G : N] = m and |N| = n. Prove that if gcd(m, n) = 1, then N is the unique subgroup of order n.
2. Let G1, G2 be groups and N1, N2 be normal subgroups of G1, G2, respectively.
a. Prove that N1 × N2 is a normal subgroup of G1 × G2.
b. Prove that (G1 × G2)/(N1 × N2) × (G1/N1) × (G2/N2).
3. Let G be a group that acts on a set S.
a. The kernal of the action defined as K = {g ∈ G | gs = s, ∀ s ∈ S}. Prove that K is a normal subgroup of G.
b. Assume that G is transitive, that is, given any x, y ∈ S, there exists h ∈ G such that hx = y. Prove that |S| = |G : Gw| for any w ∈ S.
4. Let G be a finite group and let P be a p-subgroup with NG(P) = P. Prove that P is not contained in any proper normal subgroup of G.
5. Let G be a finite group with |G| = 105. Prove that G has a normal subgroup N with N ≠ {e} and N ≠ G.

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.