(i) Determine whether ф defines a hornom Orphism. (ii) Find ker ф :-0€ G | ф(G) e} and in ф ф(G). (ii) Draw Cayley diag...
(i) Determine whether φ defines a homomorphism. (ii) Find ker ф :-(g E G I ф(G)-e) and inn ф d(G). (ii) Draw Cayley diagrams of the domain and codomain, and arrange them so one can "visually see" the cosets of ker φ in G. Draw dotted lines around these cosets. (iv) Is the quotient G/ker ф a group? If so, what is it isomorphic to? Here is an example of Step (iii) for the map o: Z6 Z3, defined by...
I have to use the following theorems to determine whether or not it is possible for the given orders to be simple. Theorem 1: |G|=1 or prime, then it is simple. Theorem 2: If |G| = (2 times an odd integer), the G is not simple. Theorem 3: n is an element of positive integers, n is not prime, p is prime, and p|n. If 1 is the only divisor of n that is congruent to 1 (mod p) then...