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Higman–Sims asymptotic formula

From Wikipedia, the free encyclopedia

In finite group theory, the Higman–Sims asymptotic formula gives an asymptotic estimate on number of groups of prime power order.

Statement

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Let be a (fixed) prime number. Define as the number of isomorphism classes of groups of order . Then:

Here, the big-O notation is with respect to , not with respect to (the constant under the big-O notation may depend on ).

References

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  • Kantor, William M. (1990). "Some topics in asymptotic group theory". Groups, Combinatorics and Geometry. Durham. pp. 403–421.
  • Higman, Graham (1960). "Enumerating p‐Groups. I: Inequalities". Proceedings of the London Mathematical Society. 3 (1): 24–30.
  • Sims, Charles C. (1965). "Enumerating p‐Groups". Proceedings of the London Mathematical Society. 3 (1): 151–166.