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Seema posted an Question
April 05, 2020 • 19:10 pm 50 points
  • CSIR NET
  • Mathematical Sciences

Solve 89 & 90

89. Let G denote the group S4 x S1. Then (A) a 2-Sylow subgroup of G is normal (B) a 3-Sylow subgroup of G is normal (C) G has a non-trivial normal subgroup (D) G has a normal sub-group of order 72 90. Let X be a normal Hausdorff space. Let A. A2, Az be closed subsets of X which are pairwise disjoint. Then there always exists a continuous real valued function f on X such that fx)=a, if x EA.i =1.2,3. (A) Iff each a. is either 0 or 1 (B) Iff at least two of the numbers aj. d2. a are equal (C) For all real values of a.as, a (D) Only if one among the setsiA. A2 and As is empty

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    Eduncle best-answer

    Dear Seema,

    Greetings!!

    Please refer the attached image for solution.

    Thank You for asking your Query.

    WhatsApp Image 2020-04-13 at 17.44.44.jpeg
  • Sourav ghosh

    If it the direct product group is S4×S1.Then order of this group is |S4|×|S1|=24×1=24. Lagrange's theorem , Order of a subgroup divides Order of the group.And the group has a subgroup( may be normal) of order 72!!!!!

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    subgroup has order 72 how is it possible

    eduncle-logo-app

    I also want to know how option d is correct. I prove that the given group has a nontrivial normal subgroup of order 3. Ask for explanation from those who says option d is correct. It may be correct but i don't know why.

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