djxl said:The order of (Z/32Z)* (the multiplicative group of units) is phi(32)=16. So if I can show that all elements have order 8 or lower, I've shown that none of them can be generators. I guess my problem is that I don't see how to do this without resorting to a calculator.
I'm not sure what you mean by the square of any element of order 8 has order 4.
To prove that a group is not cyclic, you need to show that there is no single element that can generate all the other elements in the group by repeated multiplication. This can be done by showing that the group has elements of different orders, or by proving that the group is not isomorphic to any cyclic group.
A cyclic group is a group in which all the elements can be generated by repeatedly multiplying a single element. This element is called the generator of the group and is denoted by g. A cyclic group can be finite or infinite.
Some examples of non-cyclic groups include the symmetric group Sn for n ≥ 3, the dihedral group Dn for n ≥ 3, and the alternating group An for n ≥ 5. These groups have different orders and structures, making them non-cyclic.
No, a group cannot be both cyclic and non-cyclic. This is because a group can only have one operation, and if it is cyclic, all the elements can be generated by repeated multiplication of a single element. If it is non-cyclic, there is no single element that can generate all the other elements.
Showing that a group is not cyclic is important because it helps us understand the structure and properties of the group. It also allows us to classify and categorize groups into different types, making it easier to study and compare them. Additionally, it can help us identify the subgroups and cosets of the group, which are important concepts in group theory.