(12) Where in the proof of Theorem 27.11 did we use the fact that G is an Abelian group? Why does...
(13) The operation table for Do, the dihedral group of order 12, is given in Table 27.6 Table 27.6 Operation table for D (a) Find the elements of the set Ds/Z(Ds). (b) Write the operation table for the group De/Z(D6) (c) The examples of quotient groups we have seen so far have all been Abelian groups.Is it true that every quotient group is Abelian? Explain. Give a necessary condition on a group G if G/N is a non-Abelian group. Is...
please look at red line please explain why P is normal thanks Proposition 6.4. There are (up to isomorphism) exactly three di groups of order 12: the dihedral group De, the alternating group A, and a generated by elements a,b such that lal 6, b a', and ba a-b. stinct nonabelian SKETCH OF PROOF. Verify that there is a group T of order 12 as stated (Exercise 5) and that no two of Di,A,T are isomorphic (Exercise 6). If G...