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 gHg1 = H and gKg1 = K , consequently g ( H ∩ K ) g1 ⊆ gHg1 ∩ gKg1 = H ∩ K . Since N is a normal subgroup of G if and only if N ≤ G and gNg1 ⊆ 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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 Spring '08
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 Math, Algebra, Group Theory, Normal subgroup, Coset, xh, Index of a subgroup

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