# Jordan called isomorphisms (iso.) and homomorphisms "iso. holoedriques" and "iso. meriedriques" respectively; translation of holoe/meried-driques?

Stillwell mentions in his Elements of Algebra:

The first to use the term "isomorphism" was Jordan, in his Traite des Substitutions [1870], the first textbook on group theory...Jordan used the word "isomorphism" for both isomorphisms and homomorphisms, but distinguished between the two by calling them "isomorphismes holoedriques" and "isomorphismes meriedriques" respectively.

What do the words holoedriques and meriedriques mean?

• As usual, they come from Greek : see here and this related post. Nov 22 '15 at 7:13

To quote from Jordan's Traité des substitutions et des équations algébriques (1870), p. 56 (https://archive.org/stream/traitdessubstit00jordgoog#page/n79/mode/2up):

"§ 67. A group Γ is called "isomorphic" to another group G if one can establish between their substitutions [i.e., elements] a correspondence such that: (1) each substitution of G corresponds to a single substitution of Γ, and each substitution of Γ [corresponds] to one or several substitutions of G ; (2) the product of any two substitutions of G corresponds to the product of their corresponding [substitutions in Γ ], respectively.

The isomorphism will be called "mériédrique" if several substitutions of G correspond to only one substitution of Γ, "holoédrique" in the contrary case."

Thus an "isomorphisme mériédrique" corresponds to a modern "homomorphism", whereas an "isomorphisme holoédrique" corresponds to a modern "isomorphism".

The English equivalent of the French term "holoédrique" would be the crystallographic term "holohedral" (having all of the symmetry planes that are required by the symmetry of a crystal system). The equivalent of "mériédrique" would be be the crystallographic term "merohedral" (having only a fraction of the symmetry planes that are required by the symmetry of a crystal system). Thus, in the case of "holoédrique", the correspondence between the elements of the groups Γ and G is one-to-one and thus isomorphic, whereas in the case of "mériédrique" the correspondence is not one-to-one and therefore homomorphic.

• "Isomorphisme mériédrique" means "surjective homomorphism".
– spin
Jul 29 at 13:55