The discrete Heisenberg group and its automorphism group

In this note, we give a simpler and shorter proof of the claim, which is originally due to Kahn [ 1], that the automorphism group of the discrete Heisenberg group Heis(3, Z) is isomorphic to the group (Z. Z) similar to GL(2, Z). Our method for constructing this isomorphism provides a much clearer picture of the structure of the automorphism group of Heis(3, Z). Consider the discrete Heisenberg group G = Heis(3, Z), which is the group of matrices of the form where the entries a, b, and c are elements of the group Z. The group G can also be viewed as the set of all integer triples equipped with the group law where Inn(G) similar or equal to G/C similar or equal to Z. Z is the inner automorphism group and the homomorphism. is the homomorphism Aut(G). Aut(Z. Z) induced by the homomorphism. in the exact sequence (3). Clearly, Im(.). Ker(nu).