Order of elements in finite group

[mathkid22]

Homework Statement

If G is a finite group such that all elements have prime order(other than the identity). If G has a non-trivial center then show every element has the same order.

Homework Equations

The Attempt at a Solution

Since we know that G has a center, we know that there is atleast one element that commutes with every element in G. I'm not sure how we can show that all elements have the same order from just this.

 

[mighty2000]

Hello,

Thank you for your post. This is an interesting problem to consider.

To show that every element in G has the same order, we can use the fact that the order of an element in a group is equal to the order of the subgroup generated by that element.

Since all elements in G have prime order, this means that the subgroups generated by each element must also have prime order.

Now, let's consider an arbitrary element g in G. Since g is an element of G, it must commute with all other elements in G, including the non-trivial element in the center.

This means that the subgroup generated by g must also commute with the center of G. Since the center of G is non-trivial, this means that the subgroup generated by g must contain all elements in the center of G.

Since the center of G is a subgroup of G, this means that the subgroup generated by g must be equal to the entire group G.

Therefore, the order of g is equal to the order of G, which is the same for all elements in G.

I hope this helps. Let me know if you have any further questions.