Group of order 100 with no element of order 4?

[kpoltorak]

1. Homework Statement [/]
Is there a group G with order 100 such that it has no element of order 4? How would one go about proving the existence of such a group?



2. Homework Equations [/]
For every prime divisor p of a group, there exists an element with order p.

The Attempt at a Solution

 

[jbunniii]

Do you know about direct products? If so, you should be able to construct such a group pretty easily.

 

[kpoltorak]

Yes I do have some knowledge of direct products. Could I construct a group as such: $$\mathbb{Z}_{5}\times\mathbb{Z}_{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$$?

 

[jbunniii]

Yes I do have some knowledge of direct products. Could I construct a group as such: $$\mathbb{Z}_{5}\times\mathbb{Z}_{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$$?

That's the one I had in mind.