All questions if possible but at least questions

the concepts. will be needing help in my quizzes and exams if possible all are taken through my school portal with multiple questions and fill in the blanks, let me know if you can be of help.

ersity maths). I would like some assistance as soon as possible (in the next few hours), so if you are unavailable, I would love it if another tutor could help. These are the questions that are similar to, but are not exactly the ones I am struggling with. Solving these would give me a better chance of solving my assignment. I don't know where to begin with these: Provide a non-solvable finite group G with solvable subgroups L, K, M such that G = LK = LM, M \neq K , and show that it fits the criteria. ///// Define G, a finite p -group, such that G isn't abelian. Let K \le G such that |G:K| = p , where K is abelian. Prove that there are either 1 or p + 1 such abelian subgroups, and if there are p + 1 , then the index of Z(G) in G is p^2 ///// Define N normal subgroup, G finite group, O the intersection of all maximal subgroups of G . Prove that G = ON and N \cap O is nilpotent. ///// Define p a prime number, G a finite group, K a Sylow p -subgroup of G . Assume M \le K and g^{-1}Mg \le K , where g \in G . Prove that g = km for some k \in N_G(K) (normaliser of K in G ) and some m \in C_G(M) (centraliser of K in G)