Abstract:
In mathematics, a group is a set of elements equipped with a binary operation that satisfies the axioms of identity, inverses, and associativity. A subgroup of a group is a subset which is itself a group under the same binary operation. Subgroups of a group can be displayed in a subgroup lattice, which is a diagram where lines are drawn to illustrate set containment, and which as an algebraic structure satisfies the conditions of join and meet. A normal subgroup of a group is a subgroup which is invariant under the action of conjugation. The normal subgroup lattice of a group has nice properties. For example, in the normal subgroup lattice, the join of two subgroups is simply the product of those subgroups. In 1989, A. Chermak and A. Delgado[2], discovered a new subgroup lattice for finite groups, which has since been called the Chermak-Delgado lattice. This lattice consists of those subgroups which have maximal Chermak-Delgado measure. The CD-measure is defined as the product of the order of the subgroup and the order of its centralizer subgroup. The Chermak-Delgado lattice shares many nice properties in common with the normal subgroup lattice, but it has the unique property that it is self-dual, which means that if you flip it upside down, you get the exact same diagram. Below is the CD lattice for the quaternions group, Q8.
