# Thread: How many elements of order 2 are there in this product of groups?

1. ## How many elements of order 2 are there in this product of groups?

How many elements of order 2 are there in the following abelian group of order 16? :

Z2 X Z2 X Z4

where Z2 is the integers mod 2 and Z4 is the integers mod 4.

How are these found?

Thanks for any help

2. Originally Posted by Siknature
How many elements of order 2 are there in the following abelian group of order 16? :

Z2 X Z2 X Z4

where Z2 is the integers mod 2 and Z4 is the integers mod 4.

How are these found?

Thanks for any help

In this case perhaps is easier to ask how many elements are NOT of order 2: first, note that every non-unit element is of order 2 or 4.

Now, if we agree to write the elements of the group in the form $\displaystyle (x,y,z)\,,\,\,x,y,=0,1\!\!\!\!\pmod 2\,,\,z=0,1,2,3\!\!\!\!\pmod 4$, then an element has order 4 iff it has either $\displaystyle 1\,\,\, or\,\,\, 3\!\!\!\!\pmod 4$ in the last entry...can you now count them all?

Tonio

3. Originally Posted by tonio
can you now count them all?

Tonio
Thanks, yes, now i think that this means the answer must be 7 (8 if we include the identity)