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I am currently working through all the groups with two generators, and I am up to the group with presentation $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^9 \rangle$. I have found all the finite quotients of this group, but there are also the infinite quotients of the group that I need to check. Are there any infinite quotients of this group other than the whole group? I know that there is a central element of order 2, but what I need to know is, what is this element?

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Why the downvote? The question is perfectly reasonable. – Thomas Aug 21 '13 at 4:32
    
This was answered in your earlier question. $G$ has a central element of order 2. – Derek Holt Aug 21 '13 at 6:59
    
Ok then, what I need to know is: What is the central element in terms of a and b? Also, is there a way to prove that there aren't any more quotients? – Thomas Aug 21 '13 at 7:06
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Sorry I'm in a hurry! It's the commutator $[x,y]$, where (for example) $x=b * a * b * a * b^-1 * a * b * a * b * a * b^-1 * a * b * a * b^-1 * a * b^-1 * a * b * a * b^-1 * a * b^-1 * a * b * a * b^-1 * a$, $y=b * a * b^-1 * a * b * a * b^-1 * a * b^-1 * a * b * a * b^-1 * a * b^-1 * a * b * a * b^-1 * a * b * a * b * a * b^-1 * a * b * a$. – Derek Holt Aug 21 '13 at 7:11
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"I am currently working through all the groups with two generators". Higman, Neumann and Neumann proved that every countable group embeds in a 2-generator group. So your project may take you a while... – HJRW Aug 21 '13 at 14:12

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