Question

Suppose a group G has order 35 and acts on a set S consist- ing of four elements. What can you say about the action? Hint: Must there be a fixed point?) b. 4. a. What happens if IGI-287 IGI-30?

Answer 1

A) This action induces a group homomorphismG Perm(S) of G into the group of permutations Perm(S) of S, given by o(g) (s) gs for all g E G and s S. Note that Perm(S) ~ S, so that Perm(S) 24,I K) then by first isomorphism theorem, we know that By Lagrange's theorem, we get GI/지 = Io(G) | so that | φ(G) | divides Again, since G) C Perm(S) is a subgroup, by Lagrange, 14)(G)| divides Perm(S)-24. Therefore,l(G divides 35, 24, so that l(G)l divides ged (35, 24) ,G)1, which implies K G. Thus, the action is trivial, given by gss for all g f and s ES CaieCW

B) Suppose that lG-28. The same argument as above shows that l(G)l divides 28, 24,so that lo(G)l divides gcd(28,24)4 Thus, there are three possibilities: If a) lb(G) 1 holds then the action is trivial Suppose that lG 30. The same argument 4, so as above shows that (G) divides 30, 2 that lø(G)l divides gcd(30, 24) 6. Thus, there are three possibilities: If a') lo(G)l1 holds then the action is canned with canner