abstract algebra question

abstract algebra questionNeed help with the following abstract algebra/group theory question. Please provide a detailed step by step proof. Please indicate any theorem used and do not quote any theorem that does belong this level of math. Thanks!

Suppose G is a finite group, and H is a subgroup of G.

(a) For any a ? G, we will say that a nonzero power of a visits H, when ⟨a⟩nH ̸= Ø. Every element of a ? G eventually visits the subgroup H in this way, and it should be clear to you that the power depends on the element a that we start with. Let k(a) ̸= 0, be a power depending on a such that ak(a) ? H. Show that there must, in fact, be infinitely many positive and negative integers k(a) for each a such that gk(a) ? H.

(b) Show that there is a least positive integer p(a) such that ap(a) ? H. (Thus, it follows that p defines a function on G.)

(c) Show that p(a) always divides o(a), the order of a.

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