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Classify groups that are generated by two elements $x$ and $y$ of order 2.

Could someone help me please ?

amir
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    Wow! That's a task...do you have any ideas? Have you seens Coxeter groups, dihedral groups...? – DonAntonio Nov 08 '13 at 16:05
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    Hint in the spirit of DonAntonio's comment: Think about the generators' product $xy$. Also, if you know finitely presented groups, they might help. – m_l Nov 08 '13 at 16:15
  • @DonAntonio I have seen the group that you mentionned – amir Nov 08 '13 at 16:25
  • @m_l That is the hint that I was given in the text.. – amir Nov 08 '13 at 16:34
  • @user43418 Which have you seen, dihedral groups or Coxeter groups in general? Have you seen finite presentations for dihedral groups? – m_l Nov 08 '13 at 16:40
  • @m_l I have seen dihedral groups. The only thing that I have send related to Coxeter, is the Todd-Coxeter algorithm. – amir Nov 08 '13 at 16:42
  • @m_l If you could write a complete solution, I would highly appreciate it since I have another problem of the same style and would like to have some sort of solution I can work with in order to solve it. Thank you – amir Nov 08 '13 at 16:45
  • A complete answer can be quite a feat: just think that any finite dihedral group can be written as a group generated by two involutions. Even more interesting, the infinite dihedral group is $;\langle x,y;;;x^2=y^2=1\rangle \cong C_2*C_2;$ , so even free products get into the game... – DonAntonio Nov 08 '13 at 20:04
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    This is essentially the same as... http://math.stackexchange.com/questions/160168/prove-a-group-generated-by-two-involutions-is-dihedral [An element of order 2 is called an involution.] – Bill Cook Nov 08 '13 at 21:50
  • I agree with Bill Cook. It also also a duplicate of this question (perhaps I should say "Hint: This old question gives you your answer. Can you see why?", as the fact that they are the same is a nice exercise). – user1729 Jan 16 '14 at 12:49

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This will not be a full answer, although the details I leave out shouldn't be too hard to fill in.

Consider the finitely presented group $G = \langle a, b : a^2 = b^2 = 1 \rangle$. Let $X = \langle x, y \rangle$ be a group with $x,y$ of order $2$. The groups you want to classify are the factor groups of $G$, since $$ G \rightarrow X : a \mapsto x,~ b \mapsto y$$ is a homomorphism. Thus your task is to analyse the kernel of this homomorphism. If $y(xy)^k = 1$, then $(xy)^k=y$, so $(xy)^{2k} = 1$. Similarly, if $(xy)^kx=1$, then $(xy)^{2k}=1$. This yields that if $xy$ has infinite order, then $G \cong X$. You now need to consider the groups where $xy$ has finite order, i.e. groups of the form $$ \langle a,b : a^2 = b^2 = (ab)^k = 1 \rangle $$ for some natural number $k$. (Are the generators supposed to be distinct elements? This is something you should watch out for.)

m_l
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    The OP asked for a complete solution though ;) –  Nov 08 '13 at 18:10
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    He did, but the question very much seemed like a homework problem. This is why I did not write a detailed answer that can just be copied without giving it some thought.

    That being said, my answer really doesn't leave much to be done.

    Edit: I just noted a downvote on the answer. Is there anything wrong with it?

     – m_l Nov 10 '13 at 01:30