What familiar group is the matrix 1 b (b an integer) isomorphic to?
0 1
You should phrase the question more precisely. It doesn't make sense to say that a matrix is isomorphic to a group. What would this mean anyway? What you (probably) mean is that the group generated by that matrix is isomorphic to some familiar group. Also, you should specify the operation on your matrix, which in this case is (probably) ordinary matrix multiplication.
Let A be that matrix (which I suppose is:
1 b
0 1
).
Then notice how $\displaystyle A^2$ is
1 2b
0 1,
and similarly $\displaystyle A^n$ is
1 nb
0 1.
It sure looks like we are getting the integers in this way. Try defining an isomorphism from the group generated by A to the group of integers with addition, and remember to consider the inverses (which I didn't).
thanks, ya that was a direct quote out of my workbook i guess i was supposed to be able to infer it the way you did. I was also a little tripped up on that i thought to be isomorphic the same operation had to be used (which i now understand is not the case)
thank you