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Hello everybody, like the title of this thread suggests I would like to know the solution to the following problems.

Q. 1: Given two algorithms, ( we shall notate these 'A.1' and 'A.2' ), and their respective cycle lengths, ( which shall be notated as so: 'C(A.1)' and 'C(A.2)' ), is it possible to work out the cycle length of the concatenation of the two formerly mentioned algorithms, without allowing access to a cube so as to try it out by the 'brute force method' ? ( I.e., given data that ( R U R' U ) has a cycle length of 5, and ( R U2 R' ) has a cycle length of 4, it their a strictly algebraic way of determining the cycle length of ( R U R' U R U2 R' ) ? )

Q. 2: Conversely, given one algorithm and it's cycle length is it possible to determine the cycle lengths of any smaller algorithms contained as a sub-set of the algorithm in question ? ( I.e., given the knowledge that ( R U R' U' r u r' u' ) has a cycle length of 60, is there a strictly algebraic way of working out the cycle lengths of ( R U R' U' ) and ( r u r' u' ) respectively ? )

If you are able to solve these problems I will appreciate it, or if you are unable to, at least offer helpful advice. I am aware the answer may require group theory and a fair deal of mathematics, but that is fine, I will quickly learn whatever is initially unfamiliar to me, so feel free to express your thoughts any way you wish, just be nice. Thank you.

Q. 1: Given two algorithms, ( we shall notate these 'A.1' and 'A.2' ), and their respective cycle lengths, ( which shall be notated as so: 'C(A.1)' and 'C(A.2)' ), is it possible to work out the cycle length of the concatenation of the two formerly mentioned algorithms, without allowing access to a cube so as to try it out by the 'brute force method' ? ( I.e., given data that ( R U R' U ) has a cycle length of 5, and ( R U2 R' ) has a cycle length of 4, it their a strictly algebraic way of determining the cycle length of ( R U R' U R U2 R' ) ? )

Q. 2: Conversely, given one algorithm and it's cycle length is it possible to determine the cycle lengths of any smaller algorithms contained as a sub-set of the algorithm in question ? ( I.e., given the knowledge that ( R U R' U' r u r' u' ) has a cycle length of 60, is there a strictly algebraic way of working out the cycle lengths of ( R U R' U' ) and ( r u r' u' ) respectively ? )

If you are able to solve these problems I will appreciate it, or if you are unable to, at least offer helpful advice. I am aware the answer may require group theory and a fair deal of mathematics, but that is fine, I will quickly learn whatever is initially unfamiliar to me, so feel free to express your thoughts any way you wish, just be nice. Thank you.