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Group isomorphisms

Posted this earlier but no response? Any ideas on how to do these questions?


3) a) Prove that if f : G → H is a group isomorphism, then for any element x ∈ G onehaso(x) = o(f(x)),
where o(a) denotes the order of the element a.

b) Based on the previous fact, prove that the groups C4 ad C2 × C2 are not isomorphic.
HereCn = {1, t, · · · , t^(n−1) |tn = 1}is the cyclic group of order n.

c) Prove that any group of order 4 is isomorphic either to the group C4 or C2 × C2.
Reply 1
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maths10101
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@gregorious

@poorform


any ideas guys?
Reply 2
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Zacken
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Original post by maths10101
Posted this earlier but no response? Any ideas on how to do these questions?


3) a) Prove that if f : G → H is a group isomorphism, then for any element x ∈ G onehaso(x) = o(f(x)),
where o(a) denotes the order of the element a.


Before the proper people get here, let ∣g∣=n|g| = n and ∣f(g)∣=m|f(g)| = m, then:

(f(g))n=f(gn)=f(eG)=eH(f(g))^n = f(g^n) = f(e_G) = e_H so m∣nm | n.

Then f(gm)=(f(g))m=eH=f(eG)f(g^m) = (f(g))^m = e_H = f(e_G)

use the fact that ff is injective to finish off and conclude.
Reply 3
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maths10101
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@Zacken ahh thanks mate! Helps a lot..any ideas on parts b and c?


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maths10101
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@zacken
Never mind, I got it in the end! Thanks 🤐


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Reply 5
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Zacken
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Original post by maths10101
@zacken
Never mind, I got it in the end! Thanks 🤐


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Your tags aren't working, you need to type: @Zacken and then select my name from the drop down list by manually clicking on it, otherwise it I don't get a notification.

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