While trying to read the following article

Schottenfells, Ida May. Two Non-Isomorphic Simple Groups Of The Same Order 20,160.

I found the term "holoedrically isomorphic". In an abstract for the article, I also came across the claim that "Holoedric isomorphism is the only isomorphism that can exist between two simple groups."

I haven't been able to find a definition for this term anywhere. Presumably it is a stronger notion than that of a normal isomorphism of groups? What does it mean?

  • $\begingroup$ Some googling suggests it means "simply isomorphic" aka naturally isomorphic. $\endgroup$ – anon Jul 13 '12 at 18:51

Isomorphism did not used to mean 1-1, just onto. Holoedric isomorphisms are both 1-1 and onto. See page 381 of Burnside's Theory of Groups (1ed). The standard english term was "simply isomorphic", the French term was isomorphisme holoédrique.

The corresponding term for non-injective epimorphism (so onto, not 1-1) was "multiply isomorphic" in English, and isomorphisme meriédrique in French.

"édrique" appears to be about the same as "edral" as in "dihedral".

  • Schottenfels, Ida May. "Two non-isomorphic simple groups of the same order 20,160." Ann. of Math. (2) 1 (1899/00), no. 1-4, 147–152. MR1502265 DOI:10.2307/1967281
  • $\begingroup$ Interesting. Thanks. $\endgroup$ – Dane Jul 13 '12 at 19:03

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