The Student Room Group
Uni students - Complete our survey >>
Uni students - Complete our survey >>
Uni students - Complete our survey >>

Algebra Question, Groups

Hello,
I have been studying this question for quite a while now and still I am unable to find a solution to a question such as this. It would be very much appreciated if I could have some help/assistance.
Describe explicitly the homomorphismf : D4 → S4
which identify each element of D4 with the corresponding permutation of vertices of thesquare (for example, f(H) = (14)(23), where H is the symmetry on the horizontal axis)and therefore find a subgroup of S4 which is isomorphic to D4. Same problem for the groupD5
Thanks!
Reply 1
Avatar for Zacken
Zacken
22
Original post by maths10101
Hello,
I have been studying this question for quite a while now and still I am unable to find a solution to a question such as this. It would be very much appreciated if I could have some help/assistance.
Describe explicitly the homomorphismf : D4 → S4
which identify each element of D4 with the corresponding permutation of vertices of thesquare (for example, f(H) = (14)(23), where H is the symmetry on the horizontal axis)and therefore find a subgroup of S4 which is isomorphic to D4. Same problem for the groupD5
Thanks!


It would help us help you if you could detail what you've thought/done yourself.

Edit: Take one element of D4D_4, can you see what element in S4S_4 that ff (the homomorphism) will send it to?

Example: What does the 90 degree rotation of a square do to it's vertices? So the 90 degree rotation in D4D_4 is mapped to (13)(24)(13)(24) in S4S_4. What can you say about the 270 degree rotation? What about the 180 degree rotation?
(edited 8 years ago)
Reply 2
Avatar for maths10101
maths10101
OP
5
Original post by Zacken
It would help us help you if you could detail what you've thought/done yourself.

Edit: Take one element of D4D_4, can you see what element in S4S_4 that ff (the homomorphism) will send it to?

Example: What does the 90 degree rotation of a square do to it's vertices? So the 90 degree rotation in D4D_4 is mapped to (13)(24)(13)(24) in S4S_4. What can you say about the 270 degree rotation? What about the 180 degree rotation?


It's just this one that I'm real clueless on...I'm not getting anywhere with it in the slightest :/
Original post by maths10101
It's just this one that I'm real clueless on...I'm not getting anywhere with it in the slightest :/


You can view the symmetries of a square as permutations of the vertices.

If you can describe each symmetry as a permutation, then you are done.

So in cycle notation for permutations (if you don't know this, you should learn it - makes computations far easier), the permutation (1 2 3 4) acts on a square by moving vertex 1 to where vertex 2 is, vertex 2 to vertex 3, etc. just as a quarter turn rotation does. So [quarter turn rotation in D_4] --> (1 2 3 4) in S_4.

Draw a square with its vertices labelled and just write down the permutations of the labels corresponding to symmetries of the square.
Original post by Zacken
the 90 degree rotation in D4D_4 is mapped to (13)(24)(13)(24) in S4S_4.


No it's not - that element is obviously order 2 (a product of disjoint transpositions), but the 90 degree rotation is clearly order 4! What you've written is one of the reflections or the 180 rotation. Depends if it reverses orientation or not given the labelling.
(edited 8 years ago)
Reply 5
Avatar for Zacken
Zacken
22
Original post by FireGarden
No it's not - that element is obviously order 2 (a product of disjoint transpositions), but the 90 degree rotation is clearly order 4! What you've written is one of the reflections or the 180 rotation. Depends if it reverses orientation or not given the labelling.


Urgh, silly typo on my part, I think. Thanks for pointing it out.

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you