Word problem for groups

hbghlyj

群$G$由$x,y$生成,满足关系$x^3 = y^2 = (xy)^2 = 1$,求$|G|$.
math.stackexchange.com/questions/993443/how-t … ts-x-and-y-subject-o
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:
$$1,x,x^2,y,xy,x^2y$$

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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.

Czhang271828

回复 1# hbghlyj

$a^n=b^2=(ab)^2=1$, 不就二面体群 $D_{n}$ 的定义嘛

hbghlyj

回复 2# Czhang271828
是呀.1#是从这个规则来证明$D_3$是有6个元素的.