Re: (newbie) Elementary abelian group & GF(p^n)

In article <f5044dc1-cb3f-4f24-b345-9e7687363076@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<bleezpo@xxxxxxxxx> wrote:
Hello,
let K := GF(p^n). My book says that K(+) is an elementary abelian
group of order p^n and that K* is a cyclic group of order p^n-1.

Indeed.

Consider A = C_3 x C_3, it is elementary abelian of order 9. With
abuse of notation, le me denote A as the product {1,a,a^2} x
{1,b,b^2}.

I have to find an isomorphism between A(.) and K(+).

Well, unless K = GF(3^2), you are going to have a hard time doing so,
yes?

So I assume that in fact you have K = GF(3^2).

K(+) contains 9 elements. One element is 1, the identity, and it
generates a cyclic group of order 3, namely {0, 1, 2}. Since K(+) has
nine elements, and so far you have only accounted for 3 elements,
there has to be some element x that is not in {0,1,2}. The cyclic
subgroup generated by x will have either 1, 3, or 9 elements. Cannot
have 9 elements if you already showed that K(+) is elementary
abelian. It cannot have 1 element only either (why not?). So it has 3
distinct elements, namely 0=3x, x, and 2x. Now consider {0, x, 2x} and
{0,1,2}. How many elements to they have in common?

Now consider the subgroup of K(+) generated by 1 and x. What is it?

At this point, an isomorphism should be clear.

I have no
experience about field theory, but I tried to figure it out this way:
I can write K as the direct sum {0, j, 2j} (+) {0, k, 2k}; here we
have two finite fields of order 3 that give rise

No, you do not have two finite fields of order 3. K contains only ONE
field of order 3, and it is {0, 1, 2}. You have no warrant for
asserting that the sets you have will be closed under multiplication.


with respect to the
group part, to a finite group of order 9 whose elements are

0, k, 2k, j, j+k, j+2k, 2j, 2j+k, 2(j+k).

K(+) is clearly isomorphic with A(.). Now I have problems in
identifying what K* should be...

Huh? You were asked to show that K(+) is isomorphic to A. Why are you
trying to "identify K*"? It will have nothing to do with the
structure of A, since A is not a field.


--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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