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What is the order of a group $G $ generated by two elements $x$ and $y$ subject only to the relations $x^3 = y^2 = (xy)^2 = 1$? List the subgroups of $G$.

Since the above relation is the ‘only’ relation, I presume that the order of $x$ is $3$ and the order of $y$ is $ 2$. Also y is the inverse of itself, and xy is the inverse of itself. The inverse of $x$ is $x^2$.

I have calculated the elements of the group manually using the given relation.

$$G = \{1, x, x^2,y, xy, x^2y \}$$

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I want to know if there is a formula to calculate the order of $G$? I came across another question where the above relation is $x^3 = y^2 = (xy)^3 = 1$. In this case the group $G has order 12. It is very time consuming to calculate the elements manually. Is there a better way?

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First, we have

$$

xyxy = 1 \implies yx = x^{-1}y^{-1} = x^2y

$$

Since $xy = yx^2$, we conclude that every element of $G$ can be written in the form $x^jy^k$, which means that there are at most $|x| \cdot |y| = 6$ elements.

Listing the elements of $G$, we have:

$$1,x,x^2,y,xy,x^2y$$

Elaboration, per request:

Suppose we have a word such as

$x^2 y xy$

we can reduce this with the relation $yx = x^2y$ as

$$

x^2 y^2 xy = \\

x^2 y(yx) y =\\

x^2 y x^2 y y =\\

x^2 (yx) x y^2 =\\

x^2 x^2 (y x) y^2 =\\

x^2 x^2 x^2 y y^2 =\\

x^6 y^3

$$

Perhaps you see how such a method could be extended inductively.

We have $xy=(xy)^{-1}=y^{-1}x^{-1}$, hence $yxy=x^{-1}$. It is well-known that the group

$\langle x,y \mid x^3=y^2=1, yxy=x^{-1}\rangle $ is the dihedral group $D_3$. It has $6$ elements. Moreover $D_3\simeq C_2\ltimes C_3\simeq S_3$, with $C_2=\langle y \mid y^2=1\rangle$ and $C_3=\langle x \mid x^3=1\rangle$.

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