Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

One exercise in Robinson's A Course in the Theory of Groups is to prove that the groups which have the projective property are necessarily free. I'm not able to prove that because I haven't gotten to that chapter yet, but I tried to find in the book any mention of groups that have the injective property. I know that the injective abelian groups are exactly the divisible groups. But nothing is said about injectivity in the class of all groups. Why is that?

share|improve this question
1  
The non-existence of non-trivial injective groups appeared today on MO: mathoverflow.net/questions/100245 –  t.b. Jun 21 '12 at 17:27

1 Answer 1

up vote 11 down vote accepted

"The only injective object in the category of groups is the trivial group," is the statement of the theorem in M. Nogin's "A short proof of Eilenberg and Moore's theorem," 2007. The cited work of S. Eilenberg and J.C. Moore is "Foundations of relative homological algebra," 1965.

share|improve this answer
    
Thank you very much! –  user23211 Jan 27 '12 at 0:24
2  
Or, as Mac Lane put it, "every fascist group consists only of the identity element." [1950, Duality for Groups]. –  Zhen Lin Jan 27 '12 at 0:31
    
Was that paper actually published? It basically follows exactly a hint to an exercise in D.L. Johnson's "Topics in the Theory of Group Presentations", published over 30 years ago! And the exercise is in the first 10 pages of the book! –  user641 Jan 27 '12 at 2:05
    
That's true: tinyurl.com/7rpvjrc (link to the Google Books) -- exercise 7. –  Damian Sobota Jan 27 '12 at 2:58

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.