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..

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Algebra and Number Theory Topics covered: groups. Problem Set 4 = 4.1. Let G be a group. Suppose for elements g, h E G we have (gh)" e (e is the neutral element of G) for some n. Show that in that case (hg)" = e. 4.2. Given two groups G and G2, their product is the group G x G (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 G x G2 if and only if: (1) The subgroups G1, G2 are normal. (2) Gn G = {e}. (3) G = GG2, i.e. every element g G can be represented as a product 9192 with 91 G1, 92 G2. 4.3. Let G be a finite group together with an action G x XX 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 = x 9EG where by X9 we mean the set of points of X fixed by an element g (i.e. the action by g maps any point xX 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 g E G of order p. 4.5. Let F be a finite field. Prove that the multiplicative group Fx of F is cyclic. Algebra and Number Theory Topics covered: groups. Problem Set 4 = 4.1. Let G be a group. Suppose for elements g, h E G we have (gh)" e (e is the neutral element of G) for some n. Show that in that case (hg)" = e. 4.2. Given two groups G and G2, their product is the group G x G (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 G x G2 if and only if: (1) The subgroups G1, G2 are normal. (2) Gn G = {e}. (3) G = GG2, i.e. every element g G can be represented as a product 9192 with 91 G1, 92 G2. 4.3. Let G be a finite group together with an action G x XX 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 = x 9EG where by X9 we mean the set of points of X fixed by an element g (i.e. the action by g maps any point xX 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 g E G of order p. 4.5. Let F be a finite field. Prove that the multiplicative group Fx of F is cyclic.