# Another Abstract Algebra Question

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. Im 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!

## Answers and Replies

Dick
Science Advisor
Homework Helper
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.