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Finite simple groups as expanders.

Martin Kassabov1, Alexander Lubotzky, Nikolay Nikolov

  • 1Department of Mathematics, Cornell University, Ithaca, NY 14853, USA.

Proceedings of the National Academy of Sciences of the United States of America
|April 8, 2006
PubMed
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Researchers proved that most finite simple groups can be represented as epsilon-expanders using a specific number of generators, enhancing understanding of group theory and graph theory. This finding has implications for the structure and properties of these fundamental mathematical objects.

Area of Science:

  • Group Theory
  • Graph Theory
  • Combinatorics

Background:

  • Finite simple groups are fundamental building blocks in mathematics.
  • Cayley graphs are essential tools for studying the structure of groups.
  • Expander graphs have significant applications in computer science and mathematics.

Purpose of the Study:

  • To investigate the existence of expander properties in Cayley graphs of finite simple groups.
  • To determine the number of generators required for these expander properties.
  • To exclude specific families of groups, such as Suzuki groups, from the main result.

Main Methods:

  • Utilizing advanced techniques in group theory and spectral graph theory.
  • Analyzing the properties of Cayley graphs constructed from specific sets of generators.

Related Experiment Videos

  • Applying combinatorial arguments to establish the expander conditions.

    • Main Results:

    • Existence of a constant k and epsilon for non-abelian finite simple groups (excluding Suzuki groups).
    • Demonstration that the Cayley graph Cay(G; S) with k generators satisfies the epsilon-expander property.
    • Establishing a general condition for a class of mathematical structures.

    Conclusions:

    • The study confirms that a broad class of finite simple groups exhibit expander properties.
    • This result contributes to the understanding of the geometric and combinatorial nature of finite groups.
    • The findings have potential implications for areas like random walks on groups and network design.