Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What's the automorphism group of this covering?

enter image description here

I know why this is a covering, but I don't know how to find the automorphism group of this covering.

I need help, thanks

share|improve this question

1 Answer 1

up vote 3 down vote accepted

Two automorphisms of a path-connected covering coincide iff they coincide at one point. 6 points lie above the node, thus the automorphism group can be identified as a subgroup of the permutation group on a 6 point set. Call the six points $\lbrace i_1,i_2,i_3,o_1,o_2,o_3\rbrace$ : the nodes labeled $i$ are those on the inner circle and one labeled $o$ lie on the outer circle. The lift of $b$ produces the permutation $(i_1i_2i_3)(o_1o_2o_3)$, while the lift of $a$ gives the permutation $(i_1o_1)(i_2o_2)(i_3o_3)$, so it seems that the group $G$ of the covering is isomorphic to the subgroup of $S_6$ generated by these two permutations. They commute, and so we should have $G\simeq\Bbb Z/6\Bbb Z$.

share|improve this answer
    
Thank you for your answer, only now I could understand what you said. –  user42912 Jan 27 '13 at 14:07
1  
I'm glad this has been helpful to you :) –  Olivier Bégassat Jan 27 '13 at 18:58

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.