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# feb09 - H ∩ K is a subgroup of G see Exercise 2.1.10(a on...

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Math 4124 Wednesday, February 9 February 9, Ungraded Homework Exercise 3.1.4 on page 85 Prove that in the quotient group G / N , ( gN ) α = g α N for all α Z . If α is positive, this is proved by induction. The result is certainly true if α = 0 (both sides are then the identity N ); if it is true for α = s , then we have ( gN ) s + 1 = ( gN )( gN ) s = gN ( g s N ) = g s + 1 N as required. On the other hand if α is negative, then - α is positive and we have ( gN ) α = (( gN ) - 1 ) - α = ( g - 1 N ) - α = ( g - 1 ) - α N = g α N and we are finished. Exercise 3.1.36 on page 89 Prove that if G / Z ( G ) is cyclic, then G is abelian. Write Z = Z ( G ) . If G / Z is cyclic, then G / Z = xZ for some x G . This means that the left cosets of Z in G are of the form x a Z for some a Z and z Z . Therefore every element of G is of the form x a z . Suppose we have another such element, say x b w (so w Z ). Then ( x a z )( x b w ) = zwx a x b = wzx b x a = ( wx b )( zx a ) , consequently any two elements of G commute and we have proven that G is abelian. 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
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Unformatted text preview: 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. Let G be a group and let H ≤ G . Prove that the formula g · ( xH ) = gxH for g , x ∈ G deﬁnes an action of G on the left cosets of H in G . We should note that the formula is well deﬁned because if xH = yH , then g · ( xH ) = gxH = g ( xH ) = g ( yH ) = gyH = g · ( yH ) . We now have (i) 1 · ( xH ) = 1 xH = xH . (ii) If g , k ∈ G , then g · ( k · xH ) = g · ( kxH ) = gkxH = ( gk ) · ( xH ) . This shows that we have an action, as required....
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