Re: A question in number theory!

On 2월12일, 오후10시52분, Timothy Murphy <gayle...@xxxxxxxxxx> wrote:
Cooper wrote:
Assume there is a group G whose order is p-1 and d | p-1. ----(1)

Then, is there exactly d solutions in G satisfying x^d=e?

For example, if G=(F_p)*, it is true.

This is obviously not true.
Eg S_3 has 4 elements satisfying x^2 = e,
as does C_2 x C_2.



The reason I wondering this as follows;

Consider the multiplicative character group mod (p).
Then there are exactly phi(p)=p-1 characters and if d | p-1, the book
I have says there are exactly d characters
s.t.(khai)^d =(khai)_0 (principle character).

If (1) is not true, I am wondering character group (mod p) is
isomorphic to (F_p)* and more generally, if m has primitive roots,
then charater group (mod m) is isomorphic to (Z/mZ)*?

If G is any finite abelian group,
the dual group G* (ie the group formed by the characters)
is isomorphic to G.

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland


Sorry. I admit that I did not fully think this problem since I am not
convinced the result.
But it turns out that the more thing Murphy is holds.
Thanks for everyone who replied and I will think more on the final
conclusion.
.

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