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There's a problem in topics in algebra by I.N.Herstein in the chapter of automorphism. If |G|$\geq$3(yet finite), then |Aut(G)|$\gneqq$1. Now if G is non abelian, then Z(G)$\neq$G. Thus there exists an inner automorphism which is not the identity mapping. Now, if G is abelian, then T:G$\rightarrow$G, T(g) = -g. Then T is a non trivial automorphism, iff every element of G is not of order 2. Thus the case that remains is when every element of G is of order 2. Now I can prove it using little linear algebra and direct product. But how can I get it done using only the concept of automorphism, and the isomorphism theorems?

Pierre-Guy Plamondon
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Rio
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    Why would you do it without using that such a group where every non-unit element is of order 2 is a vector space over $;\Bbb F_2\cong\Bbb Z_2;$ ? I think this is the simplest, shortest and most elegant way I can think of, and anyone learning, or attempting to, group theory must already know basic linear algebra. – DonAntonio Sep 05 '16 at 18:39
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    There exists only one group with two elements, and this has trivial automorphism group. So you need $|G| \ge 3$, otherwise the statement becomes false. – Crostul Sep 05 '16 at 18:48
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    @DonAntonio: Someone taking a first abstract algebra course has almost certainly had some sort of linear algebra course, but that course may have been heavily matrix-oriented and have dealt exclusively with vector spaces over $\Bbb R$. For such students the argument in question will require quite a bit more explanation than I think you’re envisioning. – Brian M. Scott Sep 05 '16 at 18:49
  • But when the writer is posing this problem at the moment when the reader doesn't know anything about vector space, there must be another possible way. I also think that's the simplest solution and that is why I have asked this question. – Rio Sep 05 '16 at 18:50
  • @BrianM.Scott Perhaps so, though thinking of students of mathematics, I think that wouldn't be so advanced for them. For science students of engineers yes, it most probably would. – DonAntonio Sep 05 '16 at 18:57
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    @RioDutta Brian and I were discussing that group theory is usually studied after studying some basic linear algebra – DonAntonio Sep 05 '16 at 18:57
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    @DonAntonio: I’m afraid that it would have been at the university where I taught. Everyone had essentially the same linear algebra course, and abstract algebra was often a math major’s first real theory course, or second after a ‘calculus done right’ introduction to real analysis in $\Bbb R$. And a lot of the math majors were intending to go into secondary school teaching. – Brian M. Scott Sep 05 '16 at 19:00
  • @DonAntonio I understand the general course structure. But can't we just give it some thought just for sake of curiosity? – Rio Sep 05 '16 at 19:00
  • @BrianM.Scott I see, thank you. I supose it is a matter of choices in different mathematics deparments. Here, in linear algebra I (one semester) is basically over the reals and the complex...but finite fields are also mentioned, though not much. And it has, of course, matrices, but also linear maps and functionals and etc. – DonAntonio Sep 05 '16 at 19:03
  • @RioDutta Yes, we can. I though cannot think of anything that wouldn't be completely equivalent to a linear map interchanging elements of a given ordered basis, say. And that even in the simplest case. – DonAntonio Sep 05 '16 at 19:04
  • @DonAntonio: Sounds like you have a better linear algebra course than we had. Ours was admittedly heavily influenced by the desires of the engineering school, since they supplied a large percentage of the students taking math courses. – Brian M. Scott Sep 05 '16 at 19:07
  • @BrianM.Scott Ah, that expalins, I think, a lot. Here, the basic courses for whoever studying "mathematics" are the same. That's why physicists, biologists, geologists, etc. have their own courses. In second year and later, you can choose either "extended" or "restricted" mathematics, which usually is "who goes for graduate school and etc. in mathematics and who doesn't". Topology, Measure theory, Fields extensions and Galois theory and etc. are for "extended" maths, whereas restricted (and extended, too) just take the usual diff. eq's, probability, set theory, etc. – DonAntonio Sep 05 '16 at 19:11
  • Compare with the related question. I still think that in this case we cannot describe a non-trivial automorphism with operations intrinsic to the group. – Jyrki Lahtonen Sep 05 '16 at 19:24
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    @DonAntonio: We had to keep those students, since for many years we had very few math majors. The large number of normalized student credit hours that we produced as a service department were all that persuaded the administration to let us offer real analysis and abstract algebra to the few students who signed up, and even then we could run them only in alternate years for quite a while. – Brian M. Scott Sep 05 '16 at 19:25
  • @BrianM.Scott I see. That's a small university, I gather. Although here, and I think everywhere else more or less, we also have/had some problems with mathematics majors. We even lowered the Admission Test Grade to be accepted in mathematics...but there's nothing to do: tough stuff tends to keep most of people away. – DonAntonio Sep 05 '16 at 19:29
  • I think Jirky´s answer there makes it pretty clear that, after all, trying to avoid linear algebra is almost-almost pointless, though I guess it is, strictly formally, possible. – DonAntonio Sep 05 '16 at 19:32
  • @DonAntonio: It was only ten years old when I started there; it took over the plant and much of the faculty of Fenn College. It’s always been hard to count enrolment, since CSU has always had a large number of part-time students, but the figure currently published is over $17,000$, of which I think about $12,000$ is undergraduate, and I think that it’s been on that order for quite a few years now. – Brian M. Scott Sep 05 '16 at 19:38
  • @BrianM.Scott Is CSU = Colorado State Univ.? Perhaps Cleveland State Univ., as Fenn College is in Cleveland, right? – DonAntonio Sep 05 '16 at 19:46
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    @DonAntonio: Yes, Cleveland State. The former Fenn College was a small private college in downtown Cleveland, and CSU’s Fenn College of Engineering continues its name. Fenn Tower, now a dorm but formerly the administration building, was once a luxury hotel; I once taught a course there in a fourth-floor room that had once been an indoor swimming pool! – Brian M. Scott Sep 05 '16 at 20:05

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