How could this group possibly have elements of this order?

[jdinatale]

Homework Statement

It seems to me that every element of Z_45 has order 1, 3, 5, 9, 15, or 45. It seems impossible to have an element of order 2 by Lagrange's Theorem.

Is there another way of looking at this problem?

 

[micromass]

For example, (30,0,0,0) has order 2.

Homework Statement

It seems to me that every element of Z_45 has order 1, 3, 5, 9, 15, or 45. It seems impossible to have an element of order 2 by Lagrange's Theorem.

Is there another way of looking at this problem?

 

[jdinatale]

For example, (30,0,0,0) has order 2.

I see! Do you think a counting argument would be acceptable? For example there are two elements of order 2 in Z_60, (0 and 30), one element of order 2 in Z_45 (0), two elements of order 2 in Z_12 (0 and 6) and two elements of order 2 in Z_36 (0 and 18)

So it seems to me, all of the elements of order 2 in A would be:

(0, 0, 0, 0)
(0, 0, 0, 18)
(0, 0, 6, 0)
(0, 0, 6, 18)
(30, 0, 0, 0)
(30, 0, 6, 0)
(30, 0, 0, 18)
(30, 0, 6, 18)

So 8 elements, correct?

 

[micromass]

(0,0,0,0) does not have order 2.

 

[jdinatale]

(0,0,0,0) does not have order 2.

Oh thank you, you are correct. Is my argument mathematically correct where I just list out all of the possible elements of order 2?

Also, do you know of a better way to look at the number of subgroups of index 2 in A?

 

[micromass]

Yes, the rest of what you did is absolutely correct!