Krishna Shukla posted an Question
September 24, 2021 • 13:25 pm 30 points
  • IIT JAM
  • Mathematics (MA)

Please show all homomorphic map to s₄ to z₂₄.......

please show all homomorphic map to S₄ to Z₂₄........

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  • Anonymous User best-answer

    numbers of all homomorphism from S4 to Z24 is 2

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    how does this result come??what is the explanation behind it ...?..... i have explanation but i didnt understood that if you may... Kernel can be either S₄, or A₄ or K₄ ({(1),(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)} or {e} (only normal subgroups in S₄). If {e} is made kernel, then range has to be isomorphic to S₄, but ℤ₂₄ does not have any subgroup isomorphic to S₄, so, that cannot be kernel. If K₄ is made kernel, then range has to be isomorphic to S₃, but ℤ₂₄ has no subgroup isomorphic to S₃. If A₄ is made kernel, then range has to be isomorphic to ℤ₂, and ℤ₂₄ has one subgroup isomorphic to ℤ₂, so, one map where range is isomorphic to ℤ₂. If S₄ is made kernel, then range will be trivial, and will result in trivial homomorphism. So, 1 map with trivial range, one map with order 2 range.

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    check

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    check this is proof of formula

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    check upper attachment

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    explanation

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