How many elements of order 4?

Forum for the GRE subject test in mathematics.
marco
Posts: 263
Joined: Mon Jun 11, 2012 2:27 am

How many elements of order 4?

Postby marco » Tue Sep 11, 2012 2:19 pm

One question:

Does Z_4xZ_4 have 12 elements of order 4?
Let G= {1, 7, 43, 49, 51, 57, 93, 99, 101, 107, 143, 149, 151, 157,
193, 199} under multiplication modulo 200. Does this group also have 12 elements of order 4? I only counted 10.

Could you guys help me out with this problem?

Thanks!

vonLipwig
Posts: 51
Joined: Sat Mar 17, 2012 9:58 am

Re: How many elements of order 4?

Postby vonLipwig » Tue Sep 11, 2012 3:30 pm

Yes, that first group has twelve elements of order 4.

The second group only has eight elements of order four, namely, 7, 43, 57, 93, 107, 143, 157, 193. This shows that these two groups are not the same.

From the classification of finite abelian groups, the only possibility for this second group is Z_4 x Z_2 x Z_2. (Because there are elements of order 4, no elements of order 8, the group has order 16, and it's not Z_4 x Z_4)
Last edited by vonLipwig on Tue Sep 11, 2012 8:37 pm, edited 1 time in total.

marco
Posts: 263
Joined: Mon Jun 11, 2012 2:27 am

Re: How many elements of order 4?

Postby marco » Tue Sep 11, 2012 4:14 pm

Hmm 1 has order 4? doesn't it have order one?

vonLipwig
Posts: 51
Joined: Sat Mar 17, 2012 9:58 am

Re: How many elements of order 4?

Postby vonLipwig » Tue Sep 11, 2012 8:37 pm

Whoops! I listed all of the elements which didn't have order 4 instead. I've now corrected it.

marco
Posts: 263
Joined: Mon Jun 11, 2012 2:27 am

Re: How many elements of order 4?

Postby marco » Wed Sep 12, 2012 12:10 am

I see my error now. Thanks very much!




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