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Let D12 and D6 be the dihedral groups for the corresponding b gons. Prove that D6 is a subgroup of D12. Prove that D6 is a normal subgroup of D12.
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- 45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.
- 11. Find all normal subgroups of the alternating group .18. If is a subgroup of , and is a normal subgroup of , prove that .For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H in Z18, partition Z18 into left cosets of H, and state the index [ Z18:H ] of H in Z18. H= [ 8 ] .