groups/cyclic groups

halvizo1031

I need help here: Suppose that G is a group in which every non-identity element has order two. Show that G is commutative.


Also, Consider Zn = {0,1,....,n-1}
a. show that an element k is a generator of Zn if and only if k and n are relatively prime.

b. Is every subgroup of Zn cyclic? If so, give a proof. If not, provide an example.

CompuChip

What does it mean that G is commutative?
What possible ways to prove commutativity do you know of?

For the second one, I suggest starting with the "<==" implication (i.e. assume that k and n are relatively prime and show that k generates Z_n.

halvizo1031

for the first one, we can show commutativity with a multiplication table. How else?

for the second one, i want to start with ==> and say that the order of k is n/(m,n). but how can i show it?

CompuChip

OK let's take them one at a time.

The definition of commutativity is that xy = yx for any two elements x and y.
Can you explicitly show this in the case given?