We know PGL(2, $q$) has elements of order $q+1$ or $q-1$. Suppose $k\neq 1$, $2$ divide $q+1$ or $q-1$. It is clear that PGL(2, $q$) has an elements of order $k$. I would like to know what is the number of the elements of order $k$ and how we can get it?
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$\begingroup$ You can extract the formula you seek from various sources, for instance from the webpage groupprops.subwiki.org/wiki/… Here is a standard way to get the answer. First compute the size of the conjugacy class of an order-k cyclic subgroup C of PGL(2,q), by computing the normalizer N of C in PGL(2,q) and noting that this size is [PGL(2,q):N]. Then determine the number of conjugacy classes of such subgroups C. $\endgroup$– Michael ZieveCommented Jan 11, 2013 at 17:40
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$\begingroup$ @Michael Zieve: Thank you. If possible give me more details of your answer. $\endgroup$– MartCommented Jan 11, 2013 at 18:01
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$\begingroup$ @Mart: You can find the details in many standard sources, for instance in Suzuki's group theory book, but of course you would gain the most by trying to work them out on your own. $\endgroup$– Michael ZieveCommented Jan 11, 2013 at 18:11
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2 Answers
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By the sizes of conjugacy calasses of PGL(2, q), if $k$ divides $q+1$, then the number of elements of order $k$ is $\phi (k)q(q-1)/2$ and if $k$ divides $q-1$, then the number of elements of order $k$ is $\phi (k)q(q+1)/2$.
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$\begingroup$ @Tom: I don't know you how get the number of elements of order k. $\endgroup$– MartCommented Jan 11, 2013 at 18:03
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$\begingroup$ Now I know what is the size of the conjugacy class of an order-k cyclic subgroup C, but still I don't know what is the number of conjugacy classes of such subgroups C. By Tom's answer it must be $\phi (k)/2$. But why? Can anybody help me? $\endgroup$– MartCommented Jan 11, 2013 at 20:14
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Let $C$ be a cyclic subgroup of order $k$. Then there are $|G:N_G(C)|$ subgroups conjugate to $C$. We say $|G:N_G(C)|$ is the size of conjugacy class of $C$. In a cyclic group of order $k$, there are just $\phi(k)$ elements of order $k$.
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$\begingroup$ @Mart: Sorry I have no reference on PGL at hand. But I think $1/2$ come from $|G:N_G(C)|$. $\endgroup$– Wei ZhouCommented Jan 12, 2013 at 5:43
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$\begingroup$ @Wei Zhou: By the sizes of conjugacy classes of PGL the size of the conjugacy class of an order-k cyclic subgroup $C$ must be $q(q-1)$ or $q(q+1)$. $\endgroup$– MartCommented Jan 12, 2013 at 7:35