Final answer:
The six elements of Aut(Z₂, Z₂) are the bijective homomorphisms from Z₂ x Z₂ to itself that preserve the group structure. Aut(Z₂, Z₂) is not isomorphic to Z₆ because the groups have elements of different orders.
Step-by-step explanation:
Finding Aut(Z₂, Z₂) and Determining Isomorphism
The question asks to find all six elements of Aut(Z₂, Z₂), which denotes the automorphism group of the direct product of two copies of the cyclic group with two elements, and to determine if this group is isomorphic to Z₆, the cyclic group of order six.
To find the automorphisms, one must determine the group homomorphisms from Z₂ x Z₂ to itself that are bijective. Since Z₂ x Z₂ has four elements, there are potentially 4! (24) functions from the set to itself, but only the bijective ones that preserve the group structure qualify as automorphisms. Through analysis, one would find that there are six such automorphisms.
Next, to determine if Aut(Z₂, Z₂) is isomorphic to Z₆, one would examine the structure of both groups. Since Aut(Z₂, Z₂) contains an element of order 2 (the identity automorphism) and all elements of Z₆ have an order dividing 6, we can immediately conclude that these groups are not isomorphic due to the differing order of elements within each group.