# Lattice of Subgroups of Permutation Groups

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One of the simplest and most basic of all algebraic structures is a group. A group is a set with a binary operation that satisfies four axioms: closure, associativity, the existence of an identity, and the existence of inverses. When the operation is commutative, we say that the group is Abelian (in honor of the distinguished Norwegian mathematician Niels Henrik Abel).

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Contributed by: Jaime Rangel-Mondragon (August 2012)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The graph labels do not follow a determined order and are only for the purpose of counting or to help in referring to a particular subgroup. Mouse over a vertex to see a tooltip with the expression of the permutations generating that subgroup.

For example, snapshot 1 shows the lattice of subgroups of , with itself at the top and the identity at the bottom (this last position is common to all lattices). Vertex 6 shows the list , which means that is generated by the two permutations and in cycle notation, or, in *Mathematica* notation, {Cycle[{{1,2}}], Cycle[{{2,3}}]}. In snapshot 2, we have the lattice of , of order 16 (note that you cannot deduce the order of a group from its lattice of subgroups) formed by its 19 subgroups (including itself and the identity). The subgroup labeled 6 is generated by the permutation Cycle[{{1,3,5,7},{2,4,6,8}}]. In the same lattice, the subgroup labeled 19 is generated by two permutations.

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