# Group isomorphisms

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#1
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.
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#2
gregorious

poorform

any ideas guys?
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4 years ago
#3
(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 and , then:

so .

Then

use the fact that is injective to finish off and conclude.
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#4
@Zacken ahh thanks mate! Helps a lot..any ideas on parts b and c?

Posted from TSR Mobile
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#5
@zacken
Never mind, I got it in the end! Thanks 🤐

Posted from TSR Mobile
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4 years ago
#6
(Original post by maths10101)
@zacken
Never mind, I got it in the end! Thanks 🤐

Posted from TSR Mobile
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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