1.Hello there! I'm struggling with group solubility/solvability, Sylow subgroups, and nilpotency (university maths). I would like some assistance as soon
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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)
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