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For a group represented as $\langle x,y\mid x^4,y^5,xyx^{-1}y\rangle$, how to determine its precise order? I guess I may need to use the universal property, but how to construct functions to determine size?

The question does have some steps or hints like the show it has at most 20 elements first, then construct some subjective to get its minimal order, etc.

I wonder what is the standard procedure to solve such questions

Shaun
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qwerty
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    Do you know about semidirect products? – user1729 Nov 06 '20 at 21:05
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    (To prove that it has at most 20 elements, use the third relator to prove that every element can be written in the form $x^iy^j$. Can you see why this helps?) – user1729 Nov 06 '20 at 21:07
  • See this duplicate. – Dietrich Burde Nov 06 '20 at 21:09
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    You usually need to steps: (i) find a normal form for the elements (show that every element can be written in some specific way; this gives you an upper bound for the number of elements in the group); and (ii) use the universal property to show that the group maps onto other groups of known order (this gives you a lower bound for the number of elements in the group). Combining both, you may be able to find the precise order. – Arturo Magidin Nov 06 '20 at 21:13
  • In GAP, use F:=FreeGroup(2);; Rels:=[(F.1)^4, (F.2)^5, (F.1)*(F.2)*(F.1)^(-1)*(F.2)];; Size(F/Rels); – Shaun Nov 06 '20 at 21:18
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    Incidentally, it has been proved that there is no standard procedure for solving all problems of this type - determining the order of a group given by a finite presentation is an undecidable problem. – Derek Holt Nov 07 '20 at 10:35

1 Answers1

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For instance, in this particular case, first the third "anticommutativity relation" allows you to write every element in the form $a^ib^j$. That together with the first two relations tells us that the order is at most $20$.

Next note that the elements of a certain semi-direct product $\Bbb Z_5\rtimes_\varphi\Bbb Z_4$ satisfy the given relations. Namely the one with $\varphi:\Bbb Z_4\to\rm{Aut}\Bbb Z_5\cong\Bbb Z_4$ given by $\varphi(1)=2$.

This implies that there is a surjective homomorphism from $G$ to this group. Thus we have a lower bound of $20$ for the order as well.