The continuous Lie groups of all gauge theories describe elementary particles almost completely. Furthermore, these necessarily satisfy all four relational rules or properties of those mathematical structures among elements of the common groups.
Property 1 – products of elements also belong in the groups.
Property 2 – group elements are associative.
Property 3 – an identity is a unique element of each group.
Property 4 – Each element has an inverse which is also a member of the group.
Members belonging to Hadamard Orthogonal N-dimensionally Extended Space-Time (HONEST) groups require only property 1 and 2 for their descriptions which in addition to elementary particles also include virtual particles and the quantum vacuum.


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