Let G, H and K be groups such that K is a normal subgroup of H and H is a normal subgroup of G. Then ( G/K ) / ( H/K ) = G / H. This is the Third Isomorphism Theorem or the Chain Theorem.

For example in

D4 (1)

D4 / { e, a2 } = C2 X C2 (2)

{ e, a2, a, a3 } / { a, a2 } = C2 (3)

( D4 / { e, a2 } ) / ( { e, a2, a, a3 } / { a, a2 } ) = C2 X C2 / C2 = C2 (4)

but also:

D4 / { e, a2, a, a3 } = C2 (5)

(1)

(2)

(3)

(4)

(5)

2-2018 Teaching by misleading

2 months ago

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