Isomorphism question - please help!

[havnek]

Main Question or Discussion Point

Hi everyone. Im new to these forums. I do Computer System Engineering at Brunel university in London. I did Maths and Physics at A-level but I'm struggling with some of the maths in my Engineering Maths module. Could someone please help me with the exam question I have attached with this post. I have searched high and low on the internet for advice or an algorithm on spotting/proving isomorphism but have failed to find a conclusive method. Any help at all would be really helpful, thanks!

 

[Nick M]

Think of them in 3 dimensions and rotate them. Look for the vectors to line up.

 

[havnek]

I haven't the foggiest; that is literally how the question appears in my past exam paper. There is no further information! :(

 

[mrandersdk]

Notice that the function

f(1) = 6
f(2) = 8
f(3) = 10
f(4) = 7
f(5) = 9
f(6) = 3
f(7) = 5
f(8) = 2
f(9) = 4
f(10) = 1

is a isomorphism from the pentagram into it self, that is all points are the same, if you just start at some point and start to construct it you can't go wrong. So just pick a point and begin.

I have made the labeling for you, if you can't construct it yourself, but try yourself first it's not so hard.

 

[havnek]

thank you so much that is a great help!

 

[mrandersdk]

No problem. Of cause

"is a isomorphism from the pentagram into it self, that is all points are the same, if you just start at some point and start to construct it you can't go wrong"

need some elaboration, i've only shown that the outer points are the same as the inner point, but it seem pretty obvious that all outer points are the same, and that all inner points are the same... right?

 

[havnek]

yes, but that was my problem. I was starting by labelling all my outer vertices in a clockwise order the same as the pentagon, ie 1 2 3 4 5, where as in fact (in ur example solution) it goes 1 2 3 9 6. I realise the error i was making - Once you label the first few outer vertices, trace their routes the same as the pentagon and you will find they are isomorphic. I know my exam question will not be any harder than this question so I think with my current understanding I should be fine. Thanks again.