(i) There are two non-isomorphic groups of order 4: C4 and C2xC2. Let C3 be a...

(i) There are two non-isomorphic groups of order 4: C4 and C2xC2. Let C3 be a cyclic group of order 3. For each group G of order 4, determine all possible homomorphisms f an element of Hom(C3, Aut(G)).

(ii) For C3 and each G of order 4 as above, determine all possible homomorphisms phi an element of Hom(G, Aut(C3)).

Solutions

Expert Solution

The solution is based on the simple result that, if C is a cyclic group with generator a, then any group homomorphism from C is determined by the image of a. Thus determining all the group homomorphisms is equivalent to finding all the possible images of the generator.

The problem is solved in various cases depending on what the group G is taken to be.

Colby Messinger answered 1 year ago

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