# feb14 - Math 4124 Monday, February 14 February 14, Ungraded...

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Math 4124 Monday, February 14 February 14, Ungraded Homework Exercise 3.1.22(a) on page 88 Prove that if H and K are normal subgroups of a group G , then their intersection H K is also a normal subgroup of G . We have already proved previously that H K is a subgroup of G ; see Exercise 2.1.10(a) on page 48. We now need to verify the normality condition: let g G . Then gHg - 1 = H and gKg - 1 = K , consequently g ( H K ) g - 1 gHg - 1 gKg - 1 = H K . Since N is a normal subgroup of G if and only if N G and gNg - 1 N for all g G , the result is proven. Exercise 3.1.33 on page 88 Find all normal subgroups of D 8 and for each of these ﬁnd the isomorphism type of its corresponding quotient. We shall write G = D 8 = { r , s | r 4 = s 2 = e , rs = sr - 1 } . Thus the elements of D 8 are r i and sr i for i = 0 , 1 , 2 , 3. A very useful result (which we shall cover in class) is that if H G and | G / H | = 2, then H ± G . Let

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## This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.

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feb14 - Math 4124 Monday, February 14 February 14, Ungraded...

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