Question

Modern Algebra core standard: I can verify if a given group is a normal subgroup Let G and H be two groups. Let K={(g,e)∈G×H:g∈G}. Show that K is a normal subgroup of G×H.

Answer 1

To prove that K is a normal subgroup of G×H, we need to demonstrate that the conjugate of any element in K by any element in G×H is still in K. We showed that the conjugate of any (g',e) in K by (g,h) in G×H is (gg'g^{-1}, e), which is indeed in K, verifying that K is a normal subgroup.

Step-by-step explanation:

To verify whether K is a normal subgroup of G×H where G and H are groups and K is defined as K={(g,e)∈G×H:g∈G}, we need to show that for any element (g,h) in G×H, the conjugate of any element of K by (g,h) is again an element of K. That is, for (g',e) ∈ K, we must have (g,h)(g',e)(g,h)-1 ∈ K. Since H is a group, it contains the identity element e, and since G is a group, for any g in G, there exists an inverse g-1 in G. Therefore, (g,h)(g',e)(g,h)-1 = (gg'g-1, heh-1) = (gg'g-1, e), which is an element of K because gg'g-1 is in G. Hence, K is normal in G×H.