SOLUTIONS OF SOME HOMEWORK PROBLEMS Problem set 1

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SOLUTIONS OF SOME HOMEWORK PROBLEMSMATH 114Problem set 14. LetD4denote the group of symmetries of a square. Find the order ofD4andlist all normal subgroups 8 elements:1, r, r2, r3,d1, d2, b1, b2,whereris the rotation on 90 ,d1, d2are flips about diagonals,b1, b2are flips aboutthe lines joining the centers of opposite sides of a square. LetNbe a normal subgroupofD4. Note thatd1=rd2r 1,b1=rb2r 1,d1d2=b1b2= ifd1 N, thend2 N, similarly forb1, b2. Note thatd1b1=r. Thus, ifNcontains a flip andN6=G, thenNeither containsd1, d2or containsb1, b2. LetNcontaind1andd2, thenN={1, d1, d2, r2}. In the same way ifNcontainsb1andb2,thenN={1, b1, b2, r2}. Finally, ifNdoes not contain flips, thenN={1, r, r2, r3}orN={1, r2}.

SOLUTIONS OF SOME HOMEWORK PROBLEMS MATH 114 Problem set 1 4. Let D4 denote the group of symmetries of a square. Find the order of D4 and list all normal subgroups in D4. Solution. D4 has 8 elements: 1,r,r2,r3, d 1,d2,b1,b2, where r is the rotation on 90 , d 1,d2 are flips about diagonals, b1,b2 are flips about the lines joining the centersof opposite sides of a square.

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