Question

A First Course in Abstract Algebra (7th Edition) Chapter S.17, Problem 4E

Wooden cubes of the same size are to be painted a different color on each face to make children's blocks. How many distinguishable blocks can be made if 8 colors or paint are available? Hint: X must be a set of functions from a set with 6 elements to a set with 8 elements.

Let A= (1,2,3,4,5,6) which corresponds to sides in the cube and B= (1,2,3,4,5,6,7,8) which corresponds to painting on the sides where F: Aـــــــــــ>B, then X= (F: Aـــــــــــ>B). This means that X consists of functions from the set A to the set of B.

1) Is it really ALL functions that are included in X? why?

2) How the group G acts on X? i.e. what is function gF: Aـــــــــــ>B if F: Aـــــــــــ>B is a function? I want to understand how the group G acts on such a functions set X. I hope that I get an answer from your experience with thanks.

Answer 1