Is it true that the order of this group is $14$ (because of $7\cdot2$)?
$$\langle S, T\mid S^7 = (S^4T)^4 = (ST)^3 = T^2 = 1\rangle$$
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$\begingroup$ I changed $<S,T|(S^4T)^2 = (ST)^3 = T^2 = 1>$ to: $$\langle ST\mid (S^4T)^4 = (ST)^3 = T^2=1\rangle$$ Not only the angle brackets changed, but also I used \mid rather than the vertical solidus, thereby automatically resulting in proper spacing. $\endgroup$– Michael HardyFeb 24, 2014 at 18:17
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1 Answer
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No, that group has order 168, in fact it is isomorphic to the simple group $\mathsf{PSL}(2,7)$. A good way to find the order of a group defined by a presentation is to use the (free) GAP software and follow the instructions on this page: http://www-circa.mcs.st-and.ac.uk/gapfpres.php
gap> f:= FreeGroup("s","t"); s:=f.1; t:=f.2;
gap> rels:=[ s^7, (s^4*t)^4, (s*t)^3, t^2];
gap> g:= f/rels;
gap> Size(g);
168
answered Feb 24, 2014 at 18:14