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)
2-2024 Quran and mathematics
7 months ago
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