# External Direct Product

• November 8th 2008, 01:32 PM
Juancd08
External Direct Product
I need to determine the number of elements in Z_25 direct product with Z_5.

i know that i need to count the elements with property that (lcm(|a|,|b|))= 5

and then break in to the cases that |a|=|b| = 5. the other case will be...

But what i am missing is know how choices can there be for both and b.

My book says that when |a|=|b| = 5. there are 4 choices for a and four choices of b. i dont know why.

I need help coming up with the choices for a and b. i need to know how many and what are they.
• November 8th 2008, 02:31 PM
ThePerfectHacker
Quote:

Originally Posted by Juancd08
I need to determine the number of elements in Z_25 direct product with Z_5.

By definition $\mathbb{Z}_{25}\times \mathbb{Z}_5 = \{ (a,b) | a\in \mathbb{Z}_{25}, b\in \mathbb{Z}_5 \}$. There are $25\cdot 5 = 125$ such elements.

This is Mine 11,2:):)th Post!!!
• November 8th 2008, 04:26 PM
Juancd08
how many elements of a specific order
I knew that, but i what I wanted to know is how many elements of a determined order is in Z_25 + Z_5. take the example how many elements of order 5. I know 5=lcm(a,b), but i dont know how to finish it.
• November 8th 2008, 05:02 PM
ThePerfectHacker
Quote:

Originally Posted by Juancd08
I knew that, but i what I wanted to know is how many elements of a determined order is in Z_25 + Z_5. take the example how many elements of order 5. I know 5=lcm(a,b), but i dont know how to finish it.

If $(a,b)$ has order $5$ it means: $|a|=|b|=5$ or $|a|=5,|b|=1$ or $|a|=1,|b|=5$.

That should help you solve the problem.
• November 8th 2008, 06:43 PM
Juancd08
sorry
Yeah but what i need to know is when does |a|=5. same for b.

Sorry for not asking the right question from the beggining.
• November 8th 2008, 06:47 PM
ThePerfectHacker
Quote:

Originally Posted by Juancd08
Yeah but what i need to know is when does |a|=5. same for b.

Sorry for not asking the right question from the beggining.

For $|a| = 5$ we can have $a=5,10,15,20$
For $|b| = 5$ we can have $b=1,2,3,4$