If K is a subgroup of G and N is a normal subgroup of G, prove that K/(K ? N) is

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If K is a subgroup of G and N is a normal subgroup of G, prove that K/(K ? N) is isomorphic to KN/N.

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Is this the first isomorphism theorem? If so, here is the proof. Define phi: K-->G/N by phi(k)=kN, for all k in K. Since phi is the restriction of the natural projection pi:G-->G/N, it is a group homomorphism. Recall that the set KN = {kn | k is in K, n is in N} is a subgroup of G (by Proposition 3.3.2 in my book). phi(H) = {gN in G/N | gN=kN for some k in K} = {gN in G/N | g is in KN} = KN/N Finally, ker(phi) = K intersect N, and so k intersect N is a normal subgroup of K. by the fundamental homomorphism theorem, we have phi(H) is isomorphic to K/ker(phi) Source(s): Abstract Algebra Third Edition (beachy, blair)

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If K is a subgroup of G and N is a normal subgroup of G, prove that K/(K ? N) is

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