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August 13th, 2007, 09:08 PM   #1
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groups and homo/iso/automorphisms

How many automorphisms are there of Z, and Z_n? (i.e. How many isomorphisms are there from Z to Z, and how many are there from Z_n to Z_n?)

How many homomorphisms are there from Z to G, where G is any group of order n? What's the easiest way to describe/list/write out all of them?

How many homomorphisms are there from ZxZ to S_n? (Perhaps not in general but juse show a few cases.)

Let V = Z/2 x Z/2 be Klein four group.
How many homomorphisms are there from V to S_2, S_3, and S_4, in each case?

(Z is the set of integers, Z_n = Z mod n, S = "permutations")
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August 14th, 2007, 05:47 AM   #2
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I'm assuming that an automorphism creates a bijection of a set to itself. If I'm interpreting this correctly, the number of automorphisms of Z should be Z!, which is equivalent to 2^Z=2^aleph_0=c (n! rises faster than 2^n but slower than 2^(2^n)). The number of automorphisms of Z_n should be n!.

I'm not going to try to guess at the remaining questions .
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August 14th, 2007, 06:49 AM   #3
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I suspect aptx4869 meant the group (Z, +) rather than the set Z.
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August 14th, 2007, 05:37 PM   #4
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Oops. Yes, sorry, all of them are meant to be groups so
Z was meant to be (Z, +),
Z_n was meant to be (Z_n, +_n),
G was meant to be (G, *),
S_n was meant to be (S_n, composition), etc.

Automorphisms are bijective and needs to satisfy the homomorphism property, so we need the identity element to be mapped to itself in all cases. So far I have found for:

Z: Due to the homomorphism property, we need f(1)=1 or f(1)=-1 as it is necessary for the map to be surjective since if we have |x|>|y| we must also have |f(x)| > |f(y)|.
Therefore there are only two automorphisms: f(x) = x which is the identity mapping, and f(x) = -x

Z/2: only one automorphism which is the identity mapping
Z/3: there are two automorphisms
Z/4: there are two automorphisms
Z/5: there are four automorphisms
Z/n for n>2
I think there should be 2*int{ (n-1)/2 } automorphisms, where int{.} is the integer function / floor function / removing the non-integer part......
though I haven't proved it yet, and what's more - I could be wrong about that. Correct me if I'm wrong.
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September 19th, 2007, 03:41 AM   #5
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You are correct about automorphisms of Z, but incorrect about Z/nZ.
Since 1 generates Z/nZ a homomorphism is completely determined by the image
of 1. Thus there are |Z/nZ| = n homomorphisms (by the way this answers your question about the number of homomorphisms from Z to G, there is exactly one for each element
a in G, given by k \mapsto a^k).

Since Z/nZ is finite, a homomorphism is injective iff it is surjective, so it is enough to show that it is surjective. If f(1) = a, the map is surjective iff a is relatively prime to n. Hence the number of automorphisms is phi(n) (where phi is the Euler phi-function). In case you didn't know:
phi(n) = \prod_i (p_i - 1)p^(e_i - 1),
where n = \prod_i p_i^(e_i) is the prime number decomposition of n.
As you see this is consistent with your claims about Z/nZ, n = 2...5.

Hope it helps

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