Question

In: Advanced Math

A) Suppose a group G has order 35 and acts on a set S consisting of...

A) Suppose a group G has order 35 and acts on a set S consisting of four elements. What can you say about the action?

B) What happens if |G|=28? |G|=30?

Expert Solution

A) This action induces a group homomorphism of into the group of permutations of , given by for all and . Note that , so that . If then by first isomorphism theorem, we know that

By Lagrange's theorem, we get , so that divides . Again, since is a subgroup, by Lagrange, divides . Therefore, divides , so that divides . Thus, , which implies . Thus, the action is trivial, given by for all and .

B)

Suppose that . The same argument as above shows that divides , so that divides . Thus, there are three possibilities:

If holds then the action is trivial.

Suppose that . The same argument as above shows that divides , so that divides . Thus, there are three possibilities:

If holds then the action is trivial.

Colby Messinger answered 2 years ago

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