Group theory 21: Groups of order 24

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This lecture is part of an online mathematics course on groups theory. It gives a survey of the groups of order 24, and discusses two of them (the symmetric group and the binary tetrahedral group) in more detail.
  1. Richard E Borcherds
  2. Groups of order 24
  3. binary tetrahedral group
  4. symmetric group
  5. group theory
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7:04 It should've been a normal subgroup of order 4, not 2, right?

xxMADxxSCIENTISTxx
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5:59 filling holes on the argument here: if kernel K is order 6, it has a subgroup W of order 3. [K:W]= 2, so W is normal in K, and therefore (since it is also a sylow 3-group in K) it is the only sylow 3 group in K. all subgroups of order 3 of K are sylow-3 subgroups of K, so W is unique subgroup of order 3. Consider an automorphism f of K. image of W has order 3, and is a subgroup of K, so f(W))=W. Now, for any g in G, consider f_g(x) =gxg^(-1). singe K is normal in G (because is kernel), f_g(K) is a subset of K, and f_(g^(-1)) is its inverse, also closed in K, and f_g(xz) = f_g(x) f_g(z) and f_g(x^-1) = f_g(x)^(-1). so f_g is an automorphism of K. so f_g(W)=W for every g, which is the definition of W being normal in G.

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