Please solve step by step and Do not Use chatGPT or AI tools

I know answers are available in chegg but those are perfectly wrong. Please solve if know the concept..

Number Theory Topics covered: groups. 4.1. Let G be a group. Suppose for elements g,hG we have (gh)n=e(e is the neutral element of G ) for some n. Show that in that case (hg)n=e. 4.2. Given two groups G1 and G2, their product is the group G1G2 (as a set, the Cartesian product of sets) with coordinate-wise multiplication. Suppose G is a group and G1,G2 are two subgroups of G. Prove that G is isomorphic to the product G1G2 if and only if: (1) The subgroups G1,G2 are normal. (2) G1G2={e}. (3) G=G1G2, i.e. every element gG can be represented as a product g1g2 with g1G1, g2G2. 4.3. Let G be a finite group together with an action GXX on a finite set X. Denote the set of orbits of the action by X/G. Prove the following orbit-counting formula (due to Burnside): X/G=G1gGXg, where by Xg we mean the set of points of X fixed by an element g (i.e. the action by g maps any point xXg to itself). 4.4. (Cauchy's theorem) Let G be a finite group and let p be a prime divisor of G. Then there is an element gG of order p. 4.5. Let F be a finite field. Prove that the multiplicative group Fof F is cyclic