let p be a prime. Show that

and show that

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- Dec 11th 2009, 02:21 PMmmsAutomorphism
let p be a prime. Show that

and show that - Dec 11th 2009, 02:58 PMBruno J.
In general, .

It's not trivial at all that is cyclic. It's not very difficult to show once you know it, but you are probably not expected to come up with the proof. Instead you are probably expected to use this as a fact.

The second one is false - you have . - Dec 11th 2009, 03:04 PMmms
thanks for taking the time for explaining...

but when you write the x on top of Zn you mean its the multiplicative group?

also, how could you prove that aut(Z8) is isomorphic to Z2 X Z2? - Dec 11th 2009, 04:18 PMBruno J.
Yes that is what I mean! Sorry for not being more clear.

To see that , just notice that every one of the four elements of has order 2. There is only one group of order 4 in which every element has order 2; it is the Klein 4-group. - Dec 11th 2009, 04:56 PMmms
i see.. thank you

i can't prove the first problem though :/ - Dec 11th 2009, 05:55 PMJose27
Notice that an homomorphism when is cyclic is completely determined once you know the image of a generator. Knowing this is determined by , but if is to be an automorphism then must be a generator ie. a nonzero element, but we have such elements. Now take automorphisms then if then so we can identify an automorphism with it's value at and this defines an isomorphism between

- Dec 11th 2009, 06:14 PMBruno J.
Yes, that is how one goes about showing that . However, it is showing that which is difficult (and which I doubt has to be proven by mms - it is probably to be assumed).

- Dec 11th 2009, 06:26 PMmms
thanks for your time...

i understand how you show that you have p-1 automorphisms, but i dont get hoy you define the isomorphism -.- plz help this dumb man (Itwasntme) - Dec 11th 2009, 06:31 PMBruno J.
Jose explained it, but I can explain some more.

For every , let be the automorphism (prove that it is an automorphism). Then the map defined by is the isomorphism you are looking for. (Prove it!) - Dec 11th 2009, 07:16 PMmms
i think i understand now...thank you :D