Analysis of Small Groups

  title={Analysis of Small Groups},
  author={Daniel G{\'a}tica-P{\'e}rez and Oya Aran and Dinesh Babu Jayagopi},
  booktitle={Social Signal Processing},
G −→ Aut(G/N) ≈ Sq. Let K denote the kernel of the map. Clearly K ⊂ N since each g ∈ K must in particular left translate N back to itself. Thus, since q is the smallest prime divisor of |G|, q = |G/N | | |G/K| = q · (product of primes p ≥ q). On the other hand, the first isomorphism theorem says that G/K is isomorphic to the image of G in Sq, a subgroup of Sq. Thus |G/K| | q!, so that |G/K| = q · (product of primes p < q). Comparing the two displays shows that |G/N | = |G/K|, and so the… 

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