# Sporadic Groups

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A group is a set of elements that is closed under an associative binary operation such that contains an identity element and each element in has an inverse .

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Contributed by: Ed Pegg Jr (April 2016)

Open content licensed under CC BY-NC-SA

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## Details

The generators of a group are a set of elements such that every element of the group can be expressed as a product of a finite number of generators. For example, this Demonstration has 10×10 matrices and over as generators for the 7920 elements of the Mathieu group . (The binary operation is matrix multiplication, and is the Galois field with two elements, 0 and 1.) For element 7920, the decimal sequence for the product is 85283493; the equivalent binary number is 101000101010101001010100101. Substituting for 1 and for 0 gives the product .

The 23 sporadic groups in this Demonstration are:

These three sporadic groups were judged to be too large for this Demonstration:

Matrix representations are from [1].

References

[1] R. A. Wilson, R. A. Parker, and J. N. Bray, "ATLAS of Finite Group Representations." (Apr 13, 2016) for.mat.bham.ac.uk/atlas/html/contents.html.

[2] J. Baez, "What's a Normal Subgroup?" (Apr 13, 2016) math.ucr.edu/home/baez/normal.html.

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