Another Abstract Algebra Question

[rocky926]

Another Abstract Algebra Question...

Every symmetry of the cube induces a permutation of the four diagonals connecting the opposite vertices of the cube. This yields a group homomorphism φ from the group G of symmetries of the Cube to S4 (4 is a subscript). Does φ map G onto S4? Is φ 1-1? If not, describe the symmetries in the kernel of φ. Determine the order of G.

So far for this problem I have drawn the cube and the 4 diagonals of the cube. Also I know that the cube has 24 symmetries. I am not sure however how you translate this into a group homomorphism. Does it mean that the group G of symmetries of the Cube contains 24 elements? In which case the order of G would be 24? Thanks for the help!

 

[Dick]

You said that you know that the cube has 24 symmetries. How do you know that? If you do know that, then why ask, "Does it mean that the group G of symmetries of the Cube contains 24 elements"? I ask because I don't know the answer myself, offhand.