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# Conjugacy classes and orders of matrices. Watch

1. The following are prime decompositions in :

(a) Give representatives for the conjugacy classes of elements of order dividing 16 in and give the order of each.

(b) Show that there are no elements of order 32 in .

(a) We can write the rational canonical forms as representatives.

, but since the determinant is zero, it is not in .

The first four matrices have order 16, since they satisfy so . The last two have order 8, since .

(b)We don’t know the factorization of , so we can’t check whether or not the matrices that satisfy it actually belong to . I wasn’t able to tell whether this was even reducible in to begin with, so I was kind of stuck here, and I was wondering if anybody could give me a hint...

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3. (Original post by Artus)
(b)We don’t know the factorization of ,
To start just replace x with in the factorisation of . It's somewhat messier after that.
4. (Original post by ghostwalker)
To start just replace x with in the factorisation of . It's somewhat messier after that.
Thanks

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Updated: August 23, 2013
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